Corrections to: Confidence Intervals for Linear Unbiased Estimators under Constrained Dependence

Exact examples that satisfy Assumptions 1, 2 and 3. Every edge is known and every degree is attained, so that $A^m=A^{\mathcal O}$ . The last column is the limiting coverage of the nominal 95% interval; in the second row the estimate is negative with probability tending to one.
Example	$n\operatorname{var}(\hat\beta)$	Limit of $n\widehat V_1(A^m;Z)$	Coverage
Paths of three, unit means $(1,-2,1)$	$29/15$	$19/15$	$0.887$
Paths of three, unit means $(2,-4,2)$	$29/15$	$-11/15$	–
Cycles of four, unit means $\pm3/2$	$4$	$7/4$	$0.805$

Known graph, $n=2000$ , 4000 replications. Example A with $a=\sqrt3$ and example B with $\mu=1$ ; the true value $n\operatorname{var}(\hat\beta)$ is 4 in both. Mean of $n\widehat V$ and coverage of the nominal 95% interval; a nonpositive variance estimate counts as a miss. Monte Carlo standard errors of the coverages are at most $0.008$ .
Estimator	A: $n\widehat V$	A: coverage	B: $n\widehat V$	B: coverage
$\widehat V_1(A^{\mathcal O})$ , as published	1.00	0.671	0.00	0.222
$\widehat V_1^{+}(A^{\mathcal O})$	5.99	0.984	16.00	1.000
$\widehat V_Y'$	15.00	1.000	47.99	1.000
$\widehat V_{+}$	4.00	0.954	9.00	0.997
Residual centering	–	–	4.01	0.950

Unknown graph, $n=120$ , $d_i=2$ , 1000 replications, true value 4; examples A ( $a=\sqrt3$ ) and B ( $\mu=1$ ). The published maximal estimator $\widehat V_1(A^m)$ beside $\widehat V_{\min}$ , the smaller of $\widehat V_1^{+}(A^{+})$ and $\widehat V_Y'$ . The 108 edges were drawn at random once. Coverage is computed as in the previous table. Monte Carlo standard errors of the coverages are at most $0.016$ .
Example	Known edges	$n\widehat V_1(A^m)$	Coverage	$n\widehat V_{\min}$	Coverage
A	none	14.90	1.000	14.90	1.000
A	27 of the 30 cycles	2.20	0.826	7.16	0.994
A	108 of the 120 edges	2.25	0.835	7.21	0.994
A	all	0.91	0.587	5.87	0.977
B	none	47.76	1.000	47.77	1.000
B	27 of the 30 cycles	4.22	0.842	20.14	1.000
B	108 of the 120 edges	4.32	0.885	20.24	1.000
B	all	$-0.15$	0.333	15.77	1.000

Exact values in the counterexample (the last two columns are rounded).
$n$	$E[n\widehat V_2^*(A^{\mathcal O};Z)]$	$n\operatorname{var}(\hat\beta)$	$\Pr(\|n\widehat V_2^*-n\operatorname{var}(\hat\beta)\|>1/2)$	$\Pr(n\operatorname{var}(\hat\beta)-n\widehat V_2>1/2)$
$2$	$3/2$	$1$	$0.5000$	$0.2500$
$8$	$15/8$	$1$	$0.6641$	$0.0352$
$32$	$63/32$	$1$	$0.8887$	$6.3\times10^{-6}$
$128$	$255/128$	$1$	$0.9970$	$2.6\times10^{-19}$

Exact limits for the least-squares intercept in the two block designs.
Covariate in a block	$n\operatorname{var}(\hat\beta)$	Limit of $n\widehat V_1$	Ratio
$x=(3,-1,-1,-1)$	$27/128$	$3/16$	$8/9$
$x=(0,1,-1,0)$	$47/192$	$53/192$	$53/47$

The hcv coefficient in the vignette’s regression, with a constant added to every outcome.
Constant added	Estimate	Standard error from $\widehat V_1$
$0$	$0.3809$	$0.1595$
$1$	$0.3809$	$0.5025$
$10$	$0.3809$	$3.8700$

Exact optima of the integer programs and of their linear relaxations in two examples with no known edge and every $d_i=1$ : six units with $\theta_iY_i-\hat\beta=(1,1,1,-1,-1,-1)$ , and three units.
Quantity	Integer program	Linear relaxation
Objective of (3.6), six units	2	3
Objective of (3.5), six units	4	6
$\widehat V_1$ of (3.1) at the optimum, six units	$5/18$	$1/3$
Objective of (3.8), three units	2	3

Standard errors for the vignette’s four coefficients: its column col7, and the two estimators recalculated using the package’s variance divisor $n-1$ .
Coefficient	`col7` as printed	(3.7) at the optimized graph	Closed form (3.9)
intercept	0.43915251	0.43915251	0.43917668
hcv	0.19199857	0.19199857	0.19200914
age	0.01474504	0.01474504	0.01474585
sex	0.27173666	0.27173666	0.27175162

Proposition 1, Corollary 1, the first sentence after Assumption 3, and Corollary 3, as printed (four-cycle blocks; the exact counterexample of the from-scratch check)

Proposition 1: p. 2242 (PDF p. 5); Corollary 1: p. 2245 (PDF p. 8); sentence after Assumption 3: p. 2246 (PDF p. 9); Corollary 3: p. 2248 (PDF p. 11)

Proposition 1, p. 2242 (PDF p. 5); Corollary 1, p. 2245 (PDF p. 8); sentence after Assumption 3, p. 2246 (PDF p. 9); Corollary 3, p. 2248 (PDF p. 11). The stated independent-sign four-cycle constructions satisfy Assumptions 1–3 and have a singleton compatible set, exact scaled variance 4 and mean bias -9/4, all stated finite moments, estimator limit 7/4, failure probabilities tending to one, and the exact all-alpha undercoverage formula. At alpha=0.05 its coverage limit is between 0.8045 and 0.8055. The a=3 construction has limit -5 and the stated positive-probability negative estimate. These counterexample facts refute the quoted printed assertions.

Historical component credits

Proved AronowAudit.P011.Wave1.cycle_counterexample_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The component is a concrete counterexample, not an existential negation of every printed result. The clipped interval is exactly T1(f); printed coverage is no larger, and positivity holds with probability tending to one for a=3/2. No all-integer-n extension is needed because T1 specifies n=4m.
Proved AronowAudit.P011.cycle_covariance_algebra Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The matrix called covariance in the description is HH^T for independent unit-variance signs; the theorem is its required algebra certificate rather than a standalone probabilistic model claim.
Proved AronowAudit.P011.cycle_block_moments Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.cycle_negative_realization Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the assigned deterministic value certificate; its actual positive probability is separately exported by R071-04, and the complete asymptotic assumptions are in R071-01.
Proved AronowAudit.P011.cycle_variance_formula Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.uniform_integral Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.saturated_graph_singleton Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.uniform_function_product Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.cycle_negative_probability Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:1380d6fc22639d54a946f62b9da059ebcd454e97e2dc4461b8c041629dc9b881

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed statements refuted] Proposition 1 (p. 2242): "Under Assumption 2, for any $\epsilon > 0$, $\lim_{n\to\infty}\Pr(|n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) = 0$." Corollary 1 (p. 2245): "Given Assumption 2, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) = 0$ for any $\epsilon > 0$." p. 2246: "Under homoskedasticity, the general estimator $\widehat V_1(A^m, Z)$ developed in Section 3.1 provides a conservative variance estimate." Corollary 3 (p. 2248): "Given Assumption 2, confidence intervals formed as $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V_1(A^m)}$ have at least $100(1-\alpha)\%$ coverage for $\beta$ in large $n$." [Counterexample, from the from-scratch check of the correction] For $m \ge 1$ let $n = N_n = 4m$. The vertices form $m$ blocks of four, labelled $1, 2, 3, 4$ within a block. $\mathcal G = \mathcal G_S$ is the disjoint union of the $m$ four-cycles $1 - 2 - 3 - 4 - 1$; $\mathcal G_R = \mathcal G_S$ (every edge is known); $d_i = 2$ and $\theta_i = 1$ for every $i$. For each block $b$ let $U_{b1}, U_{b2}, U_{b3}, U_{b4}$ be independent with $\Pr(U = 1) = \Pr(U = -1) = 1/2$, independent across blocks; $U_{br}$ is attached to edge $r$ of the cycle, the edges being $1 = \{1,2\}$, $2 = \{2,3\}$, $3 = \{3,4\}$, $4 = \{4,1\}$. With $a = 3/2$, the outcomes in block $b$ are $Y_{b1} = a + U_{b1} + U_{b4}$, $Y_{b2} = -a + U_{b1} + U_{b2}$, $Y_{b3} = a + U_{b2} + U_{b3}$, $Y_{b4} = -a + U_{b3} + U_{b4}$; that is, $Y_b = a s + H U_b$ with $s = (1, -1, 1, -1)^{\mathsf T}$ and $H = \begin{pmatrix}1&0&0&1\\1&1&0&0\\0&1&1&0\\0&0&1&1\end{pmatrix}$. Write $A_{\mathrm{cyc}} = \begin{pmatrix}0&1&0&1\\1&0&1&0\\0&1&0&1\\1&0&1&0\end{pmatrix}$. (a) [The hypotheses hold.] Each $Y_i$ is a function of the shocks on its two incident edges, so two disjoint vertex sets with no edge between them use disjoint sets of shocks and are independent: $\mathcal G$ is a dependency graph in the sense of Definition 1. The graphs and outcomes are nested as blocks are appended. $\mathrm{E}[\theta_i Y_i] = \pm 3/2$, $\beta = \mathrm{E}[\hat\beta] = 0$, so $\hat\beta$ is a LUE of $0$. $|\theta_i Y_i - \beta| \le 7/2$ ($c_1 = 7/2$). Every vertex has degree 2 in $\mathcal G_S$ ($c_2 = 2$). The covariance matrix of a block is $HH^{\mathsf T} = 2I + A_{\mathrm{cyc}}$, whose entries sum to 16, so $n\,\mathrm{var}(\hat\beta) = 4$ for every $m$ ($c_3 = 4$). $|\theta_i| = 1 < 2$ ($c_4 = 2$). Assumption 3 also holds: $\mathrm{var}(\theta_i Y_i) = 2$ for every $i$. (b) [The compatible set is a single matrix.] Every edge of $\mathcal G_S$ is in $\mathcal E_R$ and the forced edges use the whole allowance $d_i = 2$ at every vertex, so $\mathcal A(Z) = \{A^{\mathcal O}\}$ and $A^m = A^{\mathcal O}$ in every realization: $\widehat V_1(A^m;Z) = \widehat V_1(A^{\mathcal O};Z)$. (c) [The mean term.] In each block $\delta = a s$ and $\delta^{\mathsf T}(I + A_{\mathrm{cyc}})\delta = -4a^2 = -9$, so $b_n = -9/4$ for every $m$. (d) [Exact finite moments.] For every $n = 4m$: $\mathrm{E}[n\widehat V_1(A^{\mathcal O};Z)] = \frac{7}{4} - \frac{12}{n}$ and $\mathrm{var}(n\widehat V_1(A^{\mathcal O};Z)) = \frac{44}{n} - \frac{96}{n^2} - \frac{288}{n^3}$. (e) [Failure of consistency and of conservativeness.] $n\widehat V_1(A^{\mathcal O};Z) \to 7/4$ in probability while $n\,\mathrm{var}(\hat\beta) = 4$. Hence for every $0 < \epsilon < 9/4$, $\Pr(|n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) \to 1$ and $\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) \to 1$, contradicting Proposition 1, Corollary 1 and (the example being homoskedastic) the sentence on p. 2246. (f) [Coverage.] $\sqrt{n}\,\hat\beta$ converges in distribution to $N(0, 4)$ (independent identically distributed bounded block sums), and for every $0 < \alpha < 1$, $\lim_{m\to\infty}\Pr\left(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\max\{\widehat V_1(A^m;Z), 0\}}\right) = 2\Phi\left(z_{1-\alpha/2}\sqrt{7/16}\right) - 1 < 1 - \alpha$; for $\alpha = 0.05$ the limit is about $0.805$. This contradicts Corollary 3. (g) [Negative estimates.] With $a = 3$ in place of $3/2$ (then $c_1 = 5$; everything else unchanged) $n\widehat V_1(A^{\mathcal O};Z) \to 4 - 9 = -5$ in probability, and already at $n = 4$ the realization $U = (1,1,1,1)$, which has probability $1/16$, gives $X = (5, -1, 5, -1)$, $\hat\beta = 2$ and $\widehat V_1(A^{\mathcal O};Z) = -9/4$.

: SAP.P011a.T1WholeStatement

Theorem: SAP.P011a.T1

Statement hash: 8a1759fa701f482eae58041dea7da5dac73f5705d2ec9e87efad1863e43542c9

Statement definition: SAP.P011a.T1WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:24:39Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.cycle_negative_probability: observed used

AronowAudit.P011.saturated_graph_singleton: observed used

AronowAudit.P011.uniform_function_product: observed used

AronowAudit.P011.uniform_integral: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T2

Proposition 1, Corollary 1, the first sentence after Assumption 3, and Corollary 3, as printed (path-of-three blocks; the validators' counterexample)

Proposition 1: p. 2242 (PDF p. 5); Corollary 1: p. 2245 (PDF p. 8); sentence after Assumption 3: p. 2246 (PDF p. 9); Corollary 3: p. 2248 (PDF p. 11)

Proposition 1, p. 2242 (PDF p. 5); Corollary 1, p. 2245 (PDF p. 8); sentence after Assumption 3, p. 2246 (PDF p. 9); Corollary 3, p. 2248 (PDF p. 11). Both independent-sign path-block laws satisfy Assumptions 1–3, the bound 27/5, the displayed mean/covariance matrices, singleton compatibility, all S/L/Q polynomials and block moments, the samplewise centering formula and the exact finite oracle expectation. For t=1 the estimator tends to 19/15 instead of 29/15 and its all-alpha coverage is strictly deficient; at alpha=0.05 its coverage limit is between 0.8873623230 and 0.8873623231. For t=2 it tends to -11/15, is negative with probability tending to one and has zero clipped coverage. Both are correctly specified OLS intercept examples. These facts refute the printed assertions.

Historical component credits

Proved AronowAudit.P011.Wave1.path_counterexample_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The scope excludes separately assigned OLS and one-block certificates. Clipped coverage is the event explicitly requested by T2(e),(f), and t=1 undercoverage refutes the printed interval claim as its coverage cannot exceed clipped coverage.
Proved AronowAudit.P011.Wave1.path_block_moments_with_covariance Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The repaired statement explicitly exports the actual probability-law means and covariance, as well as the finite enumeration moments. Restricting t to {1,2} loses no case of T2.
Proved AronowAudit.P011.block_ols_representations Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; For T2 the assigned component is its correctly specified regression and OLS-intercept identity, not a separate full-rank certificate; the concrete nonzero t and positive K ensure the usual inverse is meaningful. Star conclusions are additional mathematics and are not credited to an absent T15 target here.
Proved AronowAudit.P011.saturated_graph_singleton Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f3a9931bd3f6ac7940df148fb8661d67df019a2e816b7e1272d2b1992a063f97

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed statements refuted: Proposition 1, Corollary 1, the sentence on p. 2246 and Corollary 3, as transcribed in P011-T1.] [Counterexample, from the validators] Fix $t \in \{1, 2\}$. For $K \ge 1$ let $n = N_n = 3K$. The vertices form $K$ blocks $\{a, b, c\}$; $\mathcal G = \mathcal G_S$ is the disjoint union of the $K$ paths $a - b - c$; $\mathcal G_R = \mathcal G_S$; the degrees are $(d_a, d_b, d_c) = (1, 2, 1)$; $\theta_i = 1$. In each block $U, V$ are independent with $\Pr(\cdot = 1) = \Pr(\cdot = -1) = 1/2$, independent across blocks, and $(Y_a, Y_b, Y_c) = \left(t + U,\ -2t + \tfrac35 U + \tfrac45 V,\ t + V\right)$. Write $B_0 = I + A_0 = \begin{pmatrix}1&1&0\\1&1&1\\0&1&1\end{pmatrix}$. (a) [The hypotheses hold.] $Y_a$ depends only on $U$ and $Y_c$ only on $V$, so the two leaves of a block are independent, and any two disjoint vertex sets with no path edge between them are functions of disjoint sets of signs: Definition 1 holds. The block mean vector is $(t, -2t, t)$ and the block covariance matrix is $\Sigma = \begin{pmatrix}1&3/5&0\\3/5&1&4/5\\0&4/5&1\end{pmatrix}$. $\beta = 0$ and $\mathrm{E}[\hat\beta] = 0$. Every $\mathrm{var}(\theta_i Y_i) = 1$ (Assumption 3). $|Y_i| \le 27/5$. Assumptions 1–3 hold with $c_1 = 6$, $c_2 = 2$, $c_3 = 29/15$, $c_4 = 2$; indeed $n\,\mathrm{var}(\hat\beta) = \frac{1}{3}1^{\mathsf T}\Sigma 1 = 29/15$ for every $K$. (b) [The compatible set is a single matrix.] The forced edges use every vertex's degree allowance, so $\mathcal A(Z) = \{A^{\mathcal O}\}$ and $A^m = A^{\mathcal O}$ in every realization. (c) [Block quantities.] With $S = 1^{\mathsf T}Y$, $L = 1^{\mathsf T}B_0Y$, $Q = Y^{\mathsf T}B_0Y$ for one block: $S = \tfrac85 U + \tfrac95 V$, $L = -2t + \tfrac{19}{5}U + \tfrac{22}{5}V$, $Q = \tfrac{29}{5} - 2t^2 - 2t(U + V) + \tfrac{94}{25}UV$; $\mathrm{var}(S) = 29/5$, $\mathrm{E}[SL] = 14$, $\mathrm{E}[Q] = \tfrac{29}{5} - 2t^2$, $\mathrm{var}(Q) = 8t^2 + \tfrac{8836}{625}$, $\mu^{\mathsf T}B_0\mu = -2t^2$, $1^{\mathsf T}B_01 = 7$. With $D = \hat\beta = \frac{1}{3K}\sum_k S_k$ and $\bar L_K = \frac1K\sum_k L_k$: $n\widehat V_1(A^m;Z) = n\widehat V_1(A^{\mathcal O};Z) = \frac{1}{3K}\sum_k Q_k - \frac23 D\bar L_K + \frac73 D^2$ in every realization. (d) [Exact expectation.] $\mathrm{E}[n\widehat V_1(A^{\mathcal O};Z)] = \frac{29 - 10t^2}{15} - \frac{217}{135K}$ for every $K \ge 1$. (e) [$t = 1$.] $n\widehat V_1(A^m;Z) \to 19/15$ in probability while $n\,\mathrm{var}(\hat\beta) = 29/15$; for every $0 < \epsilon < 2/3$ the probabilities that Proposition 1 and Corollary 1 say tend to zero tend to one. For every $0 < \alpha < 1$ the coverage probability of $\hat\beta \pm z_{1-\alpha/2}\sqrt{\max\{\widehat V_1(A^m;Z), 0\}}$ tends to $2\Phi\left(z_{1-\alpha/2}\sqrt{19/29}\right) - 1 < 1 - \alpha$; for $\alpha = 0.05$ the limit is $0.8873623230\ldots$. (f) [$t = 2$.] $n\widehat V_1(A^m;Z) \to -11/15$ in probability, so $\Pr(\widehat V_1(A^m;Z) < 0) \to 1$ and the interval of Corollary 3 is undefined with probability tending to one (and has limiting coverage zero if the estimate is clipped at zero). (g) [A correctly specified regression.] With one covariate $x = (t, -2t, t)$ per block and the model $\mathrm{E}[Y_i] = 0 + 1 \cdot x_i$, the covariate has sample mean zero, so the ordinary least squares intercept is $\bar Y = \hat\beta$ with $\theta_i = 1$: the example is an OLS coefficient of the kind described in Section 2.

: SAP.P011a.T2WholeStatement

Theorem: SAP.P011a.T2

Statement hash: 7439ef82ae9ebc36d414039ffb78dbbadb0a092864cc04899fc7a38425094775

Statement definition: SAP.P011a.T2WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:24:39Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.saturated_graph_singleton: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T3

Proposition 1, corrected (the limit of the oracle estimator with its mean term)

p. 2242 (PDF p. 5); the proof step replaced is on p. 2243 (PDF p. 6)

p. 2242 (PDF p. 5), replacing the proof step on p. 2243 (PDF p. 6). Under Assumption 2, without induced nesting of observed graphs, the scaled oracle converges to the scaled true variance plus the mean quadratic b_n. Its expectation remainder is bounded by 3(1+c2)²c1²/n and the sample variance converges to average marginal variance plus mean spread. Consistency is equivalent to b_n tending to zero; common means suffice. Conservativeness is equivalent to liminf b_n≥0; positive semidefiniteness suffices. The mean deviations sum to zero and |b_n|≤(1+c2)c1². The matrix I+A is positive semidefinite exactly for disjoint unions of cliques, with the stated clique-square variance formula.

Historical component credits

Proved AronowAudit.P011.revised_oracle_convergence Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The mathematical statement is faithful; the original row description calls this proved declaration a draft, which is stale status wording. No fidelity or severity verdict was changed after reading the record.
Proved AronowAudit.P011.revised_oracle_expectation Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The exact target constant and every k are exported; all covariance and integral moments are finite from Scaling rather than extra assumptions.
Proved AronowAudit.P011.revised_sample_variance Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The probability limit has a varying deterministic centre and places no common-unit-mean or homoskedasticity restriction on the family.
Proved AronowAudit.P011.oracle_mean_conditions Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Scaling implies |meanBias_k|<=(1+c2)c1^2, so the real liminf is finite and has its intended ordinary meaning. Both equivalences concern the oracle, as T3(iv),(v) require.
Proved AronowAudit.P011.clique_characterization Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Class labels C represent actual clique components; every finite equivalence relation has such natural labels, so this representation loses no graph. The sum is over attained labels and includes isolated vertices as singleton cliques.
Proved AronowAudit.P011.delta_sum_zero Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:e7230286068a019622367379bdd89e0c5f593fdad1c1785fc0f4f468b574cac0

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction and the validators' repair] Proposition 1 (revised). Under Assumption 2, with $X_i = \theta_i Y_i$, $\delta_i = \mathrm{E}[X_i] - \beta$ and $b_n = \frac1n\delta^{\mathsf T}(I + A^{\mathcal O})\delta$: (i) for any $\epsilon > 0$, $\lim_{n\to\infty}\Pr\left(\left|n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) - b_n\right| > \epsilon\right) = 0$; (ii) $\mathrm{E}[n\widehat V_1(A^{\mathcal O};Z)] = n\,\mathrm{var}(\hat\beta) + b_n + O(n^{-1})$; explicitly $\left|\mathrm{E}[n\widehat V_1(A^{\mathcal O};Z)] - n\,\mathrm{var}(\hat\beta) - b_n\right| \le \frac{3(1 + c_2)^2c_1^2}{n}$; (iii) $\hat\sigma_\theta^2 - \frac1n\sum_{i}\mathrm{var}(X_i) - \frac1n\sum_i\delta_i^2 \to 0$ in probability (the limit of the plug-in sample variance includes the mean term $\frac1n\sum_i\delta_i^2$); (iv) [consistency] the conclusion of the printed Proposition 1, $\lim_{n\to\infty}\Pr(|n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) = 0$ for any $\epsilon > 0$, holds if and only if $b_n \to 0$. A sufficient condition is $\mathrm{E}[\theta_i Y_i] = \beta$ for every $i \in \mathcal V_S$ and every $n$ (then $\delta = 0$); (v) [conservativeness of the oracle] $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^{\mathcal O};Z) > \epsilon) = 0$ for any $\epsilon > 0$ holds if and only if $\liminf_{n\to\infty} b_n \ge 0$. A sufficient condition is that $I + A^{\mathcal O}$ is positive semidefinite for every $n$; (vi) [which graphs] for the adjacency matrix $A$ of a simple graph, $I + A$ is positive semidefinite if and only if the graph is a disjoint union of cliques; in that case $\widehat V_1(A;Z) = \frac{1}{n^2}\sum_{\text{cliques } c}\left(\sum_{i \in c} e_i\right)^2$ with $e_i = \theta_i Y_i - \hat\beta$. The sentence of the printed proof "Asymptotic unbiasedness follows directly from linearity of expectations and $\mathrm{var}(\hat\beta) = O(n^{-1})$" (p. 2243) is replaced by (ii).

: SAP.P011a.T3WholeStatement

Theorem: SAP.P011a.T3

Statement hash: 3db19cb2ae92c859631f9d6f3f08d2aea10488c90b2d785a4905f8a712b12eaa

Statement definition: SAP.P011a.T3WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:24:39Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.delta_sum_zero: observed used

AronowAudit.P011.oracle_mean_conditions: observed not used

AronowAudit.P011.revised_oracle_convergence: observed not used

AronowAudit.P011.revised_oracle_expectation: observed not used

AronowAudit.P011.revised_sample_variance: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T4

The general expansion (F1) for quadratic-form estimators with a deterministic matrix, and the sufficient condition for conservativeness whatever the unit means

Replaces the argument of the proof of Proposition 1, pp. 2242–2244 (PDF pp. 5–7); not printed in the paper

Replaces the argument of the proof of Proposition 1, pp. 2242–2244 (PDF pp. 5–7); not printed in the paper. For deterministic symmetric matrices with uniformly bounded spectral norm, Assumption 2 gives both exact expectations, linear variance growth, and F1. PSD majorants give conservativeness for arbitrary unit means; nonnegative traces against admissible covariances suffice in place of the PSD difference condition, and only diagonal and true-edge entries enter those traces. The oracle matrix is PSD exactly for unions of cliques. Independent fair signs give the stated bounded-norm data-dependent counterexample. No nesting of observed graphs or monotonicity of observed sizes is required.

Historical component credits

Proved AronowAudit.P011.quadratic_expectation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The explicit L2 condition is implied by Assumption 2 because each weighted coordinate is a.s. bounded about the finite real beta; it is not an additional restriction on the target cases. Removing the other Scaling hypotheses extends the exact identity. No verdict changed after reading the record.
Proved AronowAudit.P011.deterministic_quadratic_expansion Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Scaling supplies finite moments and bounded covariance; the moving probability centre is exactly the bracket divided by n. Deterministic M and genuine Euclidean operator norm are essential and match T4. This row does not separately establish the cross-referenced clique characterization in the target narrative.
Counterexample proved AronowAudit.P011.data_dependent_expansion_failure Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Role: counterexample witness to extending the corrected deterministic F1 expansion to data-dependent M, not a refutation of a printed paper assertion. The sign law gives C=I, delta=0 and n Var(betaHat)=1, so every hypothesis of the proposed extension is met. No separate full-negation theorem is claimed.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:0ddd1f3279adee7ffffd94038c28851f350e9ccafe05d0cdf9d67910fcc41293

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Let $(M_n)$ be a deterministic sequence of symmetric $n \times n$ real matrices whose operator norms are bounded uniformly in $n$, $\|M_n\| \le K$, and let $\widehat V_{M_n} = \frac{1}{n^2}e^{\mathsf T}M_ne$ with $e_i = \theta_i Y_i - \hat\beta$. Under Assumption 2: (i) [exact, each $n$] $\mathrm{E}[e^{\mathsf T}M_ne] = \mathrm{tr}(PM_nPC) + \delta^{\mathsf T}M_n\delta = \mathrm{tr}(M_nC) + \delta^{\mathsf T}M_n\delta - \frac{2}{n}1^{\mathsf T}M_nC1 + (1^{\mathsf T}M_n1)\,\mathrm{var}(\hat\beta)$, where $P = I - \frac1n 11^{\mathsf T}$; (ii) $\mathrm{var}(e^{\mathsf T}M_ne) = O(n)$; (iii) (F1) $n\widehat V_{M_n} - n\,\mathrm{var}(\hat\beta) - \frac1n\left[\mathrm{tr}\{(M_n - I - A^{\mathcal O})C\} + \delta^{\mathsf T}M_n\delta\right] \to 0$ in probability; (iv) [sufficient condition] if in addition $M_n$ is positive semidefinite and $N_n := M_n - (I + A^{\mathcal O})$ is positive semidefinite for every $n$, then $\delta^{\mathsf T}M_n\delta \ge 0$ and $\mathrm{tr}(N_nC) \ge 0$, and therefore, for any $\epsilon > 0$ and whatever the unit means $\mathrm{E}[\theta_i Y_i]$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_{M_n} > \epsilon) = 0$. (The condition actually used on $N_n$ is $\mathrm{tr}(N_nC) \ge 0$ for every admissible covariance matrix $C$; only the diagonal entries of $N_n$ and its entries on the edges of $\mathcal G_S$ enter that trace.) The paper's choice $M_n = I + A^{\mathcal O}$ has $N_n = 0$ but is positive semidefinite only when $\mathcal G_S$ is a disjoint union of cliques (P011-T3 (vi)); (v) [the matrix must be deterministic] (iii) is false for a data-dependent matrix of bounded norm: let $X_1, \ldots, X_n$ be independent with $\Pr(X_i = 1) = \Pr(X_i = -1) = 1/2$ ($\theta_i = 1$, empty dependency graph, $C = I$, $\beta = 0$, $\delta = 0$) and $M = \frac1n XX^{\mathsf T}$, so that $\|M\| = 1$ and $\mathrm{tr}(M) = 1$. Then $n\widehat V_M = (1 - \hat\beta^2)^2 \to 1$ in probability and $n\widehat V_M - n\,\mathrm{var}(\hat\beta) \to 0$, whereas the bracket of (iii) is $\frac1n\mathrm{tr}\{(M - I)C\} = \frac{1 - n}{n} \to -1$. Graphs chosen by maximizing over the sample are therefore handled by domination in every sample (P011-T5 (c), P011-T7), not by (iii).

: SAP.P011a.T4WholeStatement

Theorem: SAP.P011a.T4

Statement hash: a046dc91c7282cf4037f08748cdb38a06194287d10f4400e712bf9557c2c5847

Statement definition: SAP.P011a.T4WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:24:39Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.deterministic_quadratic_expansion: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T5

Corollary 1, corrected for a partly known graph: the degree-corrected estimator, the closed-form bound, and their minimum (Proposition 1A of the correction, as strengthened by the check)

Replaces Corollary 1, p. 2245 (PDF p. 8), and the use of (3.1) in programs (3.5)–(3.6), pp. 2244–2245 (PDF pp. 7–8); not printed in the paper

Replaces Corollary 1, p. 2245 (PDF p. 8), and the use of (3.1) in programs (3.5)–(3.6), pp. 2244–2245 (PDF pp. 7–8); not printed in the paper. The degree-corrected estimator and Young bound satisfy all finite decompositions, PSD and nonnegativity conclusions, norm and oracle bounds, affine edge-objective and attained-optimum identities, and the constant-cap and regular-graph formulas, including zero degree. Under Assumption 2 and valid constraints the corrected oracle, sample maximizer, Young bound, and their stated minimum are conservative for arbitrary unit means, with no uniform bound on supplied degree caps. Common caps give the same conclusion for (3.9), without homoskedasticity. No nesting of observed graphs or monotonicity of observed sizes is required.

Historical component credits

Proved AronowAudit.P011.corrected_finite Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The all-compatible-A affine identity gives the optimal-value formula upon specializing to the actual maximizer of the same objective; the factor 2 correctly converts unordered program edges to ordered adjacency sums. The Assumption-2 norm bound and maximizing domination belong to corrected_asymptotic, not this distribution-free finite component.
Proved AronowAudit.P011.corrected_asymptotic Limited support; baseline fidelity: narrower; manifest: not used; not used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'P011_CorrectedAsymptoticCompletion.lean:162 exports the norm bound, maximizer domination and n-scaled conservativeness, but none of the four pointwise nonnegativity conclusions expressly promised by the row. Its constant-cap implication at :176 concludes only Conservative(V2prime), not the advertised finite equality VYoung = V2prime from T5(f); the PSD-majorant portion of the cited T5(b) is also absent. Conservative, defined at P011_Audit.lean:208 as a tail-probability limit, does not contain these finite assertions. P011_Audit.lean:2070 and :2648 establish finite ingredients separately, which confirms that the defect is missing conclusions rather than an invalid conservativeness result or an added degree-bound assumption.', 'suggested_fix': 'Add the T5(b) PSD-majorant conclusion, samplewise nonnegativity of the corrected oracle, corrected maximum, Young bound and their minimum, and the samplewise equality VYoung = V2prime when truncated caps are constant across units. These conjuncts can use corrected_matrix_majorant and corrected_finite without adding hypotheses.', 'second_review': "Of the explicitly credited T5(b),(c),(h),(i), T5(b) requires the PSD diagonal majorant difference as well as the norm bound; P011/P011_CorrectedAsymptoticCompletion.lean:162 exports the norm bound, domination and conservativeness but omits that PSD conclusion. The omission is mathematically elementary: correctedMatrix at P011/P011_Audit.lean:90 differs from oracleMatrix by diag(t_i d_i^A), and degreeCorrection at :86 is nonnegative because each positive truncated degree is an integer at least one. The description's additional samplewise nonnegativity and constant-cap equality promises concern finite material beyond these credited clauses, with equality belonging to T5(f), and should be narrowed or explicitly added. Retain a statement repair for T5(b), but classify its missing diagonal-positivity assertion as minor rather than major.", 'severity': 'minor'}
Proved AronowAudit.P011.adjacency_quadratic_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.young_bound_finite Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.quadratic_add Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.quadratic_diagonal Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.corrected_diagonal_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.corrected_matrix_majorant Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The matrix difference is the nonnegative diagonal correction by the definition of correctedMatrix; symmetry and nonnegative quadratic forms are real PSD, not a proposition assumed to mean the desired conclusion.
Proved AronowAudit.P011.ordered_pair_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.graph_laplacian_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.adjacency_upper_quadratic Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.degree_weight_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.psd_diagonal Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.psd_add Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.graph_laplacian_psd Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:ef2cba4a59598fb9b5ff102c434c6be1be7c9a9bf07b64bcd45285ae075b5c9e

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Let $d_i' = \min\{d_i, n-1\}$, $t_i = 1 - 1/d_i'$ if $d_i' > 0$ and $t_i = 0$ if $d_i' = 0$, and $e_i = \theta_i Y_i - \hat\beta$. For $A \in \mathcal A(Z)$ define the degree-corrected estimator $\widehat V_1^{+}(A;Z) = \widehat V_1(A;Z) + \frac{1}{n^2}\sum_{i \in \mathcal V_S} t_i\,d_i^A\,e_i^2$ (F2), i.e. $\widehat V_{M_A}$ with $M_A = I + A + \mathrm{diag}(t_i d_i^A)$, and the closed-form bound $\widehat V_Y'(Z) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}(1 + d_i')\,e_i^2$ (F3), i.e. $\widehat V_{I + D'}$. Let $A^{m+}$ be a maximizer of $\widehat V_1^{+}(A;Z)$ over $A \in \mathcal A(Z)$. Assume valid constraints ($\mathcal E_R \subseteq \mathcal E_S$, $d_i \ge \sum_j A^{\mathcal O}_{ij}$). Finite statements, in every sample and for every $n$: (a) for every $A \in \mathcal A(Z)$, $M_A = (D_A + A) + \mathrm{diag}(1 - d_i^A/d_i')$ (the ratio read as 0 when $d_i' = 0$), $x^{\mathsf T}(D_A + A)x = \sum_{i<j}A_{ij}(x_i + x_j)^2$ and $d_i^A \le d_i'$; hence $M_A$ is positive semidefinite and $\widehat V_1^{+}(A;Z) \ge 0$; (b) $M_{A^{\mathcal O}} - (I + A^{\mathcal O}) = \mathrm{diag}(t_i d_i^{A^{\mathcal O}})$ is positive semidefinite, and $\|M_{A^{\mathcal O}}\| \le 1 + 2c_2$ under Assumption 2; (c) $\widehat V_1^{+}(A^{m+};Z) \ge \widehat V_1^{+}(A^{\mathcal O};Z)$; (d) $\widehat V_1^{+}(A;Z)$ is affine in $A$: with the weights $\hat v_{ij} = e_ie_j + \frac12\left(t_ie_i^2 + t_je_j^2\right)$, $i < j$, in program (3.6), and $w^\ast$ its optimal value, $n^2\widehat V_1^{+}(A^{m+};Z) = \sum_i e_i^2 + 2w^\ast$; (e) $I + D'$ is positive semidefinite; $(I + D') - (I + A^{\mathcal O}) = (D' - D_{A^{\mathcal O}}) + (D_{A^{\mathcal O}} - A^{\mathcal O})$ with both terms positive semidefinite, where $x^{\mathsf T}(D_{A^{\mathcal O}} - A^{\mathcal O})x = \sum_{i<j}A^{\mathcal O}_{ij}(x_i - x_j)^2$; and $\widehat V_Y'(Z) \ge \widehat V_1(A;Z)$ for every $A \in \mathcal A(Z)$ (because $2e_ie_j \le e_i^2 + e_j^2$); (f) if $d_i' = d'$ for every $i$, then $\widehat V_Y'(Z) = \widehat V_2'(Z) = \hat\sigma_\theta^2(1 + d')/n$; (g) if $A$ is $d$-regular and $d_i' = D \ge 1$ for every $i$, then $\widehat V_1^{+}(A;Z) = \widehat V_1(A;Z) + d\left(1 - \frac1D\right)\frac{1}{n^2}\sum_i e_i^2$; in particular the added term is $(d - 1)\frac{1}{n^2}\sum_i e_i^2$ when $D = d \ge 1$, and $\widehat V_1^{+} = \widehat V_1$ when $d = 0$. Asymptotic statement: (h) under Assumption 2 and valid constraints, and for any unit means $\mathrm{E}[\theta_i Y_i]$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V(Z) > \epsilon) = 0$ for any $\epsilon > 0$, where $\widehat V$ is any of $\widehat V_1^{+}(A^{\mathcal O};Z)$, $\widehat V_1^{+}(A^{m+};Z)$, $\widehat V_Y'(Z)$ and $\min\{\widehat V_1^{+}(A^{m+};Z), \widehat V_Y'(Z)\}$. No bound on the supplied degrees $d_i$ is needed beyond validity: Assumption 2's bound $c_2$ on the true degrees in $\mathcal G_S$ suffices; (i) consequently, when every unit has the same degree bound $d'$, the paper's closed-form $\widehat V_2'(Z)$ of (3.9) satisfies the conclusion of (h) without Assumption 3.

: SAP.P011a.T5WholeStatement

Theorem: SAP.P011a.T5

Statement hash: abfada6a9255d43057027c73c378ba9560cf99d7a431c667366c80aa7a3deca9

Statement definition: SAP.P011a.T5WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:24:39Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.adjacency_quadratic_bound: observed used

AronowAudit.P011.adjacency_upper_quadratic: observed used

AronowAudit.P011.corrected_asymptotic: observed not used

AronowAudit.P011.corrected_diagonal_bounds: observed used

AronowAudit.P011.degree_weight_sum: observed used

AronowAudit.P011.graph_laplacian_identity: observed used

AronowAudit.P011.graph_laplacian_psd: observed used

AronowAudit.P011.ordered_pair_sum: observed used

AronowAudit.P011.psd_add: observed used

AronowAudit.P011.psd_diagonal: observed used

AronowAudit.P011.quadratic_add: observed used

AronowAudit.P011.quadratic_diagonal: observed used

AronowAudit.P011.young_bound_finite: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T6

Proposition 1, corrected for a known graph: the positive-part estimator (F4)

Replaces the use of $\widehat V_1(A^{\mathcal O};Z)$ when the dependency graph is known, p. 2242 (PDF p. 5); not printed in the paper

replaces the known-graph use of (3.1), p. 2242 (PDF p. 5). Spectral clipping is a PSD majorant with the degree norm bound, dominates both the oracle estimate and zero, and is asymptotically conservative under Assumption 2. It is the unique nearest PSD matrix in Frobenius norm, but is neither a least Loewner majorant nor affine in the adjacency entries. The four-cycle spectrum, quadratic identities, alternative majorant, exact expectation 4-12/n and limit 4 are included.

Historical component credits

Proved AronowAudit.P011.positive_part_finite Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Exports minimum distance and all finite bounds in its designated scope. Uniqueness and nonaffinity are separately credited rows R073-08 and R073-09.
Proved AronowAudit.P011.positive_part_conservative Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The probability law on real coordinate space represents arbitrary real outcome laws; bounded weighted outcomes guarantee the required moments. The public correction summary accurately states this n-scaled result.
Proved AronowAudit.P011.cycle_positive_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The designated finite algebraic component is exact, with no restriction to centered or realizable vectors.
Proved AronowAudit.P011.Wave1.positive_part_cycle_claims_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Faithful to the expressly assigned majorant and expectation/limit component. The true-variance identity remains a target-level coverage gap, not a new hypothesis on this row.
Proved AronowAudit.P011.cycle_spectral_certificate Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; As a certificate, the explicit orthogonal eigendecomposition and clipped candidate give the target eigenvalues and alternating eigenvector. It makes no independent asymptotic claim.
Proved AronowAudit.P011.cycle_majorant_certificate Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.cycle_clipped_entries Withdrawn support; baseline fidelity: wrong_credit; manifest: used; used in Solution; Excluded: final_status is wrong_credit; no independent target credit is taken.
Proved AronowAudit.P011.positive_part_nearest_unique Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the usual uniqueness characterization once the separate minimum-distance result is available.
Proved AronowAudit.P011.positive_part_objective_nonaffine Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; A single fixed-data nonaffinity witness suffices to rule out a generally affine objective. The row does not claim that every data vector gives a nonaffine function.
Proved AronowAudit.P011.positive_part_psd Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.positive_part_of_psd Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.variance_form_oracle Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:e0ca4df4fe59b2d805ffb6126e925640cf4e6836e9f713ca2bce584b60e26e11

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Suppose $A^{\mathcal O}$ is known. Let $I + A^{\mathcal O} = \sum_k\lambda_kq_kq_k^{\mathsf T}$ be an orthonormal eigen-decomposition, $(I + A^{\mathcal O})_+ = \sum_k\max(\lambda_k, 0)\,q_kq_k^{\mathsf T}$ its positive part, and $\widehat V_P(Z) = \frac{1}{n^2}e^{\mathsf T}(I + A^{\mathcal O})_+e$ with $e_i = \theta_i Y_i - \hat\beta$. (a) [finite] $(I + A^{\mathcal O})_+$ is positive semidefinite; $(I + A^{\mathcal O})_+ - (I + A^{\mathcal O}) = \sum_k\max(-\lambda_k, 0)\,q_kq_k^{\mathsf T}$ is positive semidefinite; $\|(I + A^{\mathcal O})_+\| \le 1 + \max_i\sum_j A^{\mathcal O}_{ij}$; hence in every sample $\widehat V_P(Z) \ge \max\{\widehat V_1(A^{\mathcal O};Z), 0\}$. (b) [asymptotic] Under Assumption 2, for any $\epsilon > 0$ and any unit means $\mathrm{E}[\theta_i Y_i]$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_P(Z) > \epsilon) = 0$. (c) [what is not claimed] $(I + A^{\mathcal O})_+$ is the positive semidefinite matrix nearest to $I + A^{\mathcal O}$ in Frobenius norm, but it is not a least positive semidefinite majorant in the Loewner order: for the four-cycle of P011-T1 and $\delta = (0, 1, -1, 0)$, $\delta^{\mathsf T}(I + A_{\mathrm{cyc}})_+\delta = 1$, while there is a symmetric $M$ with $M$ and $M - (I + A_{\mathrm{cyc}})$ both positive semidefinite and $\delta^{\mathsf T}M\delta = 3/5$. $\widehat V_P$ is not affine in $A$ and cannot serve as the linear objective of (3.6). (d) [the four-cycle] For $A_{\mathrm{cyc}}$ of P011-T1 the eigenvalues of $I + A_{\mathrm{cyc}}$ are $3, 1, 1, -1$, the eigenvalue $-1$ having eigenvector $\frac12(1, -1, 1, -1)$, and for all real $x$: $x^{\mathsf T}(I + A_{\mathrm{cyc}})_+x = x^{\mathsf T}(I + A_{\mathrm{cyc}})x + \frac14(x_1 - x_2 + x_3 - x_4)^2 = \frac34(x_1 + x_2 + x_3 + x_4)^2 + \frac12\left((x_1 - x_3)^2 + (x_2 - x_4)^2\right)$. In the example of P011-T1, $\mathrm{E}[n\widehat V_P(Z)] = 4 - \frac{12}{n}$ and $n\widehat V_P(Z) \to 4 = n\,\mathrm{var}(\hat\beta)$ in probability.

: SAP.P011a.T6WholeStatement

Theorem: SAP.P011a.T6

Statement hash: f96a3126befff3c9bc03e70d84afc2d19742ce9ab3334636172715465cdcac42

Statement definition: SAP.P011a.T6WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:26:42Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.cycle_clipped_entries: observed used

AronowAudit.P011.cycle_majorant_certificate: observed used

AronowAudit.P011.positive_part_conservative: observed not used

AronowAudit.P011.positive_part_of_psd: observed used

AronowAudit.P011.positive_part_psd: observed used

AronowAudit.P011.variance_form_oracle: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T7

Corollaries 1 and 3 in the degree-only case: the maximized $\widehat V_1$ when no edges are known (Proposition 1B of the correction, as strengthened by the check)

Corollary 1, p. 2245 (PDF p. 8); the degree-only case is the one described on p. 2248 (PDF p. 11)

Corollary 1, p. 2245 (PDF p. 8), and the degree-only case on p. 2248 (PDF p. 11). The degree identity and both fixed-cap finite comparisons hold. Under Assumption 2, the degree-only maximum is conservative and has conservative clipped Wald coverage, even for unbounded supplied caps; residuals are bounded by 2c1 almost surely. For independent fair signs with caps n-1, every nonconstant sample has exactly the two residual-sign cliques as optimizer, while the exact estimates converge unscaled to 1/2 and 1, respectively.

Historical component credits

Proved AronowAudit.P011.Wave1.degree_only_finite_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The simultaneous maximizers exclude no degree/data instance. D=0 gives the intended exact equalities rather than an inconsistent or negative-bound premise.
Proved AronowAudit.P011.Wave1.degree_only_conservative_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Empty recruitment is part of the corrected T7(c) claim itself. Supplied bounds may grow arbitrarily; no selector measurability premise or common-unit-mean restriction is added.
Proved AronowAudit.P011.Wave1.unbounded_caps_gap_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Faithful to the assigned values/limits component. The theorem does not export the separate assertion that every optimizer has precisely the two-clique structure when both signs occur; that is a target coverage gap.
Proved AronowAudit.P011.degree_only_edge_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; All graph types, ordered versus unordered sums, and natural-to-real degree coercions agree with the finite identity.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f89a4b0edb5f51abc2d222257be17de3a8f604df8dba0968792c01d99bd262eb

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Suppose no edges are known, $\mathcal E_R = \emptyset$, so that $\mathcal A(Z)$ is the set of binary symmetric zero-diagonal $A$ with $\sum_j A_{ij} \le d_i$ for every $i$, and let $A^m \in \mathcal A_1^m$ maximize $\widehat V_1(A;Z)$ over $\mathcal A(Z)$. Write $e_i = \theta_i Y_i - \hat\beta$, $B = \max_i|e_i|$, $d_i' = \min\{d_i, n-1\}$ and $\widehat V_Y'(Z) = \frac{1}{n^2}\sum_i(1 + d_i')e_i^2$ (P011-T5). (a) [finite identity] For every binary symmetric zero-diagonal $A$: $e^{\mathsf T}Ae = \sum_i d_i^A e_i^2 - \sum_{i<j}A_{ij}(e_i - e_j)^2$. (b) [finite bound] If $d_i' \le D$ for every $i$, with $D$ a nonnegative integer, then in every sample $0 \le n\widehat V_Y'(Z) - n\widehat V_1(A^m;Z) \le \frac{C_DB^2}{n}$, where $C_D = \sum_{d=1}^{D}(3d^2 + 2d)$. (c) [asymptotic] Under Assumption 2, with $\mathcal E_R = \emptyset$ and valid degree bounds ($d_i \ge \sum_j A^{\mathcal O}_{ij}$; the supplied $d_i$ need not be bounded in $n$), for any $\epsilon > 0$ and any unit means $\mathrm{E}[\theta_i Y_i]$: $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) = 0$. So Corollary 1, and with it Corollary 3 (in the form of P011-T8), hold as printed when only degree information is used. (d) [the equivalence in (b) needs bounded supplied degrees] Let $X_1, \ldots, X_n$ be independent with $\Pr(X_i = 1) = \Pr(X_i = -1) = 1/2$, $\theta_i = 1$, empty dependency graph (Assumption 2 holds with $c_3 = 1$), $\mathcal E_R = \emptyset$ and supplied bounds $d_i = n - 1$. Then, on the event that $\hat\beta \ne \pm 1$, the maximizing graph joins all units with positive $e_i$ into one clique and all units with negative $e_i$ into another, and $\widehat V_1(A^m;Z) = \frac12(1 - \hat\beta^2)^2 \to \frac12$ while $\widehat V_Y'(Z) = 1 - \hat\beta^2 \to 1$ in probability (unscaled values). (e) [the same comparison for the degree-corrected estimator] If $d_i' \le D$ for every $i$ and $\mathcal E_R = \emptyset$, then in every sample $0 \le \frac1n\sum_i\max(1, 2d_i')e_i^2 - n\widehat V_1^{+}(A^{m+};Z) \le \frac{B^2}{n}\sum_{d=1}^{D}(4d^2 + d)$.

: SAP.P011a.T7WholeStatement

Theorem: SAP.P011a.T7

Statement hash: b7b6cf004adf664b4e1b92484d58e0cfbfdbc319fc61cb7d4fe3bde630899d5e

Statement definition: SAP.P011a.T7WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:26:42Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.Wave1.degree_only_conservative_proof: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T8

Corollaries 1 and 3, corrected

Corollary 1: p. 2245 (PDF p. 8); Corollary 3: p. 2248 (PDF p. 11); the sentence on p. 2246 (PDF p. 9) is deleted

Corollary 1, p. 2245 (PDF p. 8), and Corollary 3, p. 2248 (PDF p. 11); the homoskedasticity sentence on p. 2246 (PDF p. 9) is deleted. Under Assumption 2, every estimator in the corrected list has one-sided asymptotic variance control and conservative clipped Wald coverage: the four corrected/Young estimators under valid constraints, spectral clipping, the degree-only original maximum, and the oracle or original maximum when the mean-bias limit inferior is nonnegative. Equal unit means and disjoint clique graphs imply that condition. The first two branches are nonnegative in every sample. The homoskedastic cycle and path counterexamples show why no such conclusion holds for the unrestricted original estimator with known edges.

Historical component credits

Proved AronowAudit.P011.Wave1.revised_corollaries_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; CorrectedClass enumerates explicit estimators and their target-prescribed admissibility conditions; it does not contain Conservative or ConservativeCoverage as fields. The quantile and clipping agree with corrected T8, and Scaling bounds meanBias, so its real liminf has the ordinary bounded-sequence meaning. The row proves the positive revised corollary pair, not the cross-referenced failures of the original printed corollaries.
Proved AronowAudit.P011.clipped_truncation Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:60921a9f5c84d6465d399dd172e1ac86aa8f24ee66e298bf1ca4d962de4524c2

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction and the validators' repair] Let $\mathcal C$ be the following list of variance estimators, each with the condition under which it is on the list: (1) $\widehat V_1^{+}(A^{\mathcal O};Z)$, $\widehat V_1^{+}(A^{m+};Z)$, $\widehat V_Y'(Z)$ and $\min\{\widehat V_1^{+}(A^{m+};Z), \widehat V_Y'(Z)\}$ of P011-T5, under valid constraints; (2) $\widehat V_P(Z)$ of P011-T6, when $A^{\mathcal O}$ is known; (3) $\widehat V_1(A^m;Z)$ with $A^m \in \mathcal A_1^m$, when $\mathcal E_R = \emptyset$ and the degree bounds are valid (P011-T7); (4) $\widehat V_1(A^{\mathcal O};Z)$ and $\widehat V_1(A^m;Z)$ with $A^m \in \mathcal A_1^m$, under valid constraints, when $\liminf_{n\to\infty}\frac1n\delta^{\mathsf T}(I + A^{\mathcal O})\delta \ge 0$, in particular when $\mathrm{E}[\theta_i Y_i] = \beta$ for every $i \in \mathcal V_S$, or when $\mathcal G_S$ is a disjoint union of cliques for every $n$. Corollary 1 (revised). Given Assumption 2, for every $\widehat V \in \mathcal C$ and any $\epsilon > 0$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V(Z) > \epsilon) = 0$. Corollary 3 (revised). Given Assumption 2, for every $\widehat V \in \mathcal C$ and every $0 < \alpha < 1$, $\liminf_{n\to\infty}\Pr\left(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\max\{\widehat V(Z), 0\}}\right) \ge 1 - \alpha$. The sentence on p. 2246, "Under homoskedasticity, the general estimator $\widehat V_1(A^m, Z)$ developed in Section 3.1 provides a conservative variance estimate", is deleted: Assumption 3 equalizes variances, not means, and the examples of P011-T1 and P011-T2 are homoskedastic. For $\widehat V_1(A^m;Z)$ with known edges and no condition on the unit means, neither conclusion holds in general (P011-T1, P011-T2).

: SAP.P011a.T8WholeStatement

Theorem: SAP.P011a.T8

Statement hash: f1b3586957bca4b4ea12fdd90741857e1b42f549b4f1dbf03268351c5070436f

Statement definition: SAP.P011a.T8WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:26:42Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.Wave1.revised_corollaries_proof: observed not used

AronowAudit.P011.oracle_mean_conditions: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T9

The residual-centred version of $\widehat V_1$ when a mean model is available, with the regularity it needs

Replaces the centring at $\hat\beta$ in (3.1) and (3.5), pp. 2242 and 2244 (PDF pp. 5 and 7), for the regression example of Section 2, pp. 2240–2241 (PDF pp. 3–4); not printed in the paper

replaces centring in (3.1) and (3.5), pp. 2242 and 2244 (PDF pp. 5 and 7), for the regression example, pp. 2240–2241 (PDF pp. 3–4); corrected statement, not printed. Under Assumption 2 and the correct bounded finite-dimensional mean model, root-n coefficient estimation gives zero-mean residuals with covariance C, the three stated stochastic norm rates, oracle consistency and maximal-estimator conservativeness. The finite symmetric quadratic perturbation bound holds. Bounded residuals and prediction error o_p(sqrt(n)) also suffice; a misspecified model adds the quadratic mean term. The independent-sign OLS example satisfies Assumption 2 with exact scaled variance 2, but has intercept variance n/2+1/(2n), residual expectation n+2-7/n (17/4 and 73/8 at n=4 and 8) and fails consistency, with the explicit slope weights and samplewise residual sum. No observed-graph nesting is required.

Historical component credits

Proved AronowAudit.P011.residual_perturbation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The statement matches the assigned finite auxiliary component; no hidden result assumption or problematic totalization is used.
Proved AronowAudit.P011.Wave1.residual_root_n_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; No OLS identity, full rank or design nonsingularity is assumed: the target allows any estimator with the stated root-n rate.
Proved AronowAudit.P011.Wave1.residual_prediction_stability_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; MeanModel integrability and estimator measurability make the target’s finite means and random estimator explicit; neither adds an exceptional-case exclusion.
Proved AronowAudit.P011.Wave1.misspecified_residual_limit_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Only T9(e) is credited. No limit or nonnegativity of its moving bias is assumed; raw zero-weight coordinates need not be bounded.
Proved AronowAudit.P011.Wave1.residual_missing_regularity_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is a counterexample to the residual correction without regularity, not a refutation of a printed residual theorem: the target itself is a corrected statement. The actual witness has n Var(betaHat)=2 identically, so its failure to approach 2 is also failure to track the true scaled variance.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:6bf48273b9966f978231e8756007846c0c7e1c4480d3c08794005e38c740f855

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Suppose in addition that $\mathrm{E}[Y_i] = x_i^{\mathsf T}\gamma$ for covariates $x_i \in \mathbb R^p$ with $p$ fixed, that the $x_i$ and the $\theta_i$ are bounded uniformly in $i$ and $n$, and that $\hat\gamma$ is an estimator with $\sqrt n\,(\hat\gamma - \gamma) = O_p(1)$. Let $z_i = \theta_i(Y_i - x_i^{\mathsf T}\gamma)$ and $r_i = \theta_i(Y_i - x_i^{\mathsf T}\hat\gamma) = z_i - \theta_i x_i^{\mathsf T}(\hat\gamma - \gamma)$, and define $\widehat V_1^{\mathrm{res}}(A;Z) = \frac{1}{n^2}r^{\mathsf T}(I + A)r$. (a) The $z_i$ have mean zero and covariance matrix $C$; $\|r - z\|_\infty = O_p(n^{-1/2})$, $\|r - z\|_2 = O_p(1)$ and $\|z\|_2 = O_p(\sqrt n)$. (b) [finite inequality] For every symmetric $M$: $|r^{\mathsf T}Mr - z^{\mathsf T}Mz| \le \|M\|\,\|r - z\|_2\left(2\|z\|_2 + \|r - z\|_2\right)$. (c) Under Assumption 2: $n\widehat V_1^{\mathrm{res}}(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) \to 0$ in probability; and for a maximizer $A^{m,\mathrm{res}}$ of $\widehat V_1^{\mathrm{res}}(A;Z)$ over $\mathcal A(Z)$ (valid constraints), $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1^{\mathrm{res}}(A^{m,\mathrm{res}};Z) > \epsilon) = 0$ for any $\epsilon > 0$. More generally the conclusions hold whenever the weighted prediction error satisfies $\|r - z\|_2 = o_p(\sqrt n)$ and the summands $z_i$ are uniformly bounded. (d) [the regularity is not implied by the paper's assumptions] Let $n = 4m$, regressors $(1, w_i)$ with $w = (1, 0, -1, 0)$ repeated, $R_i$ independent with $\Pr(R_i = 1) = \Pr(R_i = -1) = 1/2$, $\mathrm{E}[Y_i] = 0$, $Y_i = R_i$ if $w_i \ne 0$ and $Y_i = nR_i$ if $w_i = 0$. The OLS slope has weights $\theta_i = 2w_i$, so $X_i = 2w_iY_i$ is bounded by 2, the dependency graph is empty, $n\,\mathrm{var}(\hat\beta) = 2$ and $|\theta_i| < 3$: Assumption 2 holds for the slope and the linear mean model is exactly correct. Yet the fitted intercept $\bar Y$ has variance $\frac n2 + \frac{1}{2n}$, and with OLS residuals $\mathrm{E}[n\widehat V_1^{\mathrm{res}}(A^{\mathcal O};Z)] = n + 2 - \frac7n$ ($= 17/4$ at $n = 4$ and $73/8$ at $n = 8$), where $n\widehat V_1^{\mathrm{res}}(A^{\mathcal O};Z) = \frac4n\sum_{i: w_i \ne 0}(Y_i - \bar Y - w_i\hat b)^2$ and $\hat b = \frac2n\sum_i w_iY_i$; the estimate is not consistent for 2. (e) [a wrong mean model] If there is a deterministic $\gamma_\ast$ such that, with $r_i^\ast = \theta_i(Y_i - x_i^{\mathsf T}\gamma_\ast)$, $\|r - r^\ast\|_2 = o_p(\sqrt n)$ and the $r_i^\ast$ are uniformly bounded, then $n\widehat V_1^{\mathrm{res}}(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) - \frac1n\eta^{\mathsf T}(I + A^{\mathcal O})\eta \to 0$ in probability, with $\eta_i = \mathrm{E}[X_i] - \theta_i x_i^{\mathsf T}\gamma_\ast$.

: SAP.P011a.T9WholeStatement

Theorem: SAP.P011a.T9

Statement hash: b82e1d6635d6f9629df84c79403ce0fee6cf2dc5595e31b0c53bbc9fe4646961

Statement definition: SAP.P011a.T9WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:26:42Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.Wave1.misspecified_residual_limit_proof: observed not used

AronowAudit.P011.Wave1.residual_prediction_stability_proof: observed not used

AronowAudit.P011.Wave1.residual_root_n_proof: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T10

Why consistency cannot be kept for arbitrary unit means: no estimator is consistent at the $n$ -scaled level over all bounded mean patterns

Bears on the abstract, p. 2238 (PDF p. 1) ("We establish the consistency of an oracle variance estimator when a dependency graph is known") and Proposition 1, p. 2242 (PDF p. 5); not printed in the paper

bears on the abstract, p. 2238 (PDF p. 1), and Proposition 1, p. 2242 (PDF p. 5); corrected statement, not printed. For every a,b>0 the independent fixed-sign-mean and random-sign-mixture models satisfy all scaling assumptions, have the stated finite means and scaled variances b² and a²+b², and obey the exact finite mixture identity. No statistic is scaled consistent in all these models, and mixture conservativeness prevents consistency in every fixed mean pattern. Consequently no law-independent observed-data variance rule is universally scaled consistent. The zero statistic is unscaled consistent under Assumption 2. Extra sampling-design information can permit unbiased variance estimation: independent half-probability inclusion of population values 1 gives betaHat=average(2I) and the unbiased estimator average(2I)/n.

Historical component credits

Proved AronowAudit.P011.Wave1.universal_consistency_impossible_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The quantifier over one fixed infinite pattern precedes the limit, and the same statistic is used under every law. The impossibility over all admissible models follows by restriction to these subfamilies, whose ancillary data agree.
Proved AronowAudit.P011.mixture_finite_moments Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The broader parameter range is mathematically harmless. Finite support ensures genuine moments, so integral defaults do not cause these equalities.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:ec5726c269cc1fd10cee7c85691406d0e48d586b29ea166b21478365372a2d8f

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the from-scratch check of the correction] Fix $a > 0$ and $b > 0$, and let $\varepsilon_1, \varepsilon_2, \ldots$ be independent with $\Pr(\varepsilon_i = 1) = \Pr(\varepsilon_i = -1) = 1/2$. For each deterministic sequence $s = (s_1, s_2, \ldots) \in \{-1, 1\}^{\mathbb N}$ let $P_s$ be the model $X_i = a s_i + b\varepsilon_i$ with $\theta_i = 1$ and the empty dependency graph. Under $P_s$: $\mathrm{E}[X_i] = a s_i$, $\beta = \beta_n = \frac an\sum_{i \le n}s_i$, $|X_i - \beta_n| \le 2a + b$, and $n\,\mathrm{var}(\hat\beta) = b^2$ for every $n$; Assumption 2 holds with $c_1 = 2a + b$, $c_3 = b^2$. Let $\bar P$ be the model in which $s_1, s_2, \ldots$ are themselves independent with $\Pr(s_i = 1) = \Pr(s_i = -1) = 1/2$ and independent of the $\varepsilon_i$: under $\bar P$ the $X_i$ are independent with mean 0, $\beta = 0$, and $n\,\mathrm{var}(\hat\beta) = a^2 + b^2$; Assumption 2 holds with $c_1 = a + b$, $c_3 = a^2 + b^2$. (i) For each $n$, the law of $(X_1, \ldots, X_n)$ under $\bar P$ is the average over $s \in \{-1, 1\}^n$, with weights $2^{-n}$, of its laws under $P_s$. (ii) There is no sequence of statistics $T_n = T_n(X_1, \ldots, X_n)$ such that $T_n \to b^2$ in probability under $P_s$ for every deterministic $s$ and $T_n \to a^2 + b^2$ in probability under $\bar P$. Consequently no variance estimator $\widehat V$ computed from the observed data satisfies $n\widehat V - n\,\mathrm{var}(\hat\beta) \to 0$ in probability for every sequence of models that satisfies the paper's assumptions. (iii) If $T_n$ is conservative under $\bar P$, i.e. $\Pr(a^2 + b^2 - T_n > \epsilon) \to 0$ for every $\epsilon > 0$, then it is not true that $T_n \to b^2$ in probability under $P_s$ for every $s$: a guarantee that holds for all mean patterns costs width on some of them. (iv) [what is not claimed] Unscaled consistency for $\mathrm{var}(\hat\beta)$ is trivial (the statistic 0 has it, because $\mathrm{var}(\hat\beta) = O(1/n)$); and with additional structure, such as known sampling probabilities in a design-based setting, an unbiased variance estimator can exist.

: SAP.P011a.T10WholeStatement

Theorem: SAP.P011a.T10

Statement hash: b2cd1ca099c3942473c96db6f16a9deb0963c64b0c9a4a8f8f64ea0544556853

Statement definition: SAP.P011a.T10WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:26:42Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T11

Proof of Proposition 2, the consistency claimed for the auxiliary oracle estimator $\widehat V_2^*(A^{\mathcal O};Z)$ , as printed

p. 2246 (PDF p. 9), proof of Proposition 2

p. 2246 (PDF p. 9), proof of Proposition 2. The alternating-means counterexample refutes the quoted auxiliary proof step both as printed with < epsilon and under the intended > epsilon consistency reading. It establishes the exact finite moments and both failure limits, while preserving Proposition 2's one-sided conservativeness conclusion.

Historical component credits

Proved AronowAudit.P011.Wave1.alternating_counterexample_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This witness meets both the printed less-than event and the intended greater-than event tests at the specified positive thresholds. It also proves the valid one-sided Proposition 2 conclusion.
Proved AronowAudit.P011.Wave1.alternating_finite_moments_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is a finite supporting component of a counterexample, not a standalone asymptotic refutation; R074-06 and R074-08 supply the remaining components.
Proved AronowAudit.P011.auxiliary_empty_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The general modelling paragraph describes the project’s asymptotic framework, whereas this row is only the exact finite identity and requires none of those extra asymptotic conditions.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:471240a2600486077cef93ef434e6bbdf52c544a9f6892d5d2d2c1de156ad965

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed proof step refuted, p. 2246] "As in the proof of Proposition 1, $\hat\sigma_\theta^2$ converges in mean square. By Assumption 2, $1 \le 1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij} \le 1 + c_2$, allowing us to invoke Slutsky's Theorem and Chebyshev's Inequality to show $\lim_{n\to\infty}\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| < \epsilon) = 0$." The step is refuted both as printed and in the intended reading with $> \epsilon$ (consistency of $n\widehat V_2^*$ for $n\,\mathrm{var}(\hat\beta)$). [Counterexample, from the validator] Let $\varepsilon_1, \varepsilon_2, \ldots$ be independent with $\Pr(\varepsilon_i = 1) = \Pr(\varepsilon_i = -1) = 1/2$; $\mu_i = 2$ for odd $i$ and $\mu_i = 0$ for even $i$; $Y_i = \mu_i + \varepsilon_i$. At sample size $n$ put $p = \lceil n/2\rceil$, $r = p - \lfloor n/2\rfloor \in \{0, 1\}$, $t_n = n/(2p)$, $\theta_i = t_n$ for $i = 1, \ldots, n$, $X_i = t_nY_i$, $\beta = 1$. $\mathcal G$ is the edgeless graph on $\{1, \ldots, n\}$, every vertex is observed ($N_n = n$), $\mathcal E_R = \emptyset$, $A^{\mathcal O} = 0$, $d_i = 0$. (a) [The hypotheses hold.] $\mathrm{E}[\hat\beta] = t_n \cdot 2p/n = 1 = \beta$ for every $n$. $1/2 \le t_n \le 1$ and $|X_i - 1| \le 2$ ($c_1 = 2$). All degrees are 0 ($c_2 = 1$). $\mathrm{var}(X_i) = t_n^2 =: \sigma_n^2$ for every $i$ (Assumption 3). $n\,\mathrm{var}(\hat\beta) = t_n^2 \to 1$ ($c_3 = 1$). $|\theta_i| \le 1 < 2$ ($c_4 = 2$). The raw outcomes and the graphs are nested. (b) Because $A^{\mathcal O} = 0$, $n\widehat V_2^*(A^{\mathcal O};Z) = n\widehat V_2(A^{\mathcal O};Z) = \hat\sigma_\theta^2$, whatever the correlations. (c) [Exact decomposition.] With $\delta_i = t_n\mu_i - 1$, $M_n = \frac1n\sum_i\delta_i^2 = t_n^2\left(1 - \frac{r^2}{n^2}\right)$, $S_n = \sum_i\varepsilon_i$ and $H_n = \sum_i\delta_i\varepsilon_i$: $\hat\sigma_\theta^2 = \sigma_n^2 + M_n + \frac{2t_nH_n}{n} - \frac{t_n^2S_n^2}{n^2}$. With $R_n = \hat\sigma_\theta^2 - (\sigma_n^2 + M_n)$: $\mathrm{E}[R_n] = -\frac{\sigma_n^2}{n}$ and $\mathrm{E}[R_n^2] = \frac{4\sigma_n^2M_n}{n} + \sigma_n^4\left(\frac{3}{n^2} - \frac{2}{n^3}\right)$. (d) [The limit.] $D_n := n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) = M_n + R_n \to 1$ in probability. For even $n$: $t_n = 1$, $M_n = 1$, $n\,\mathrm{var}(\hat\beta) = 1$, $n\widehat V_2^* = 2 + \frac{2T_n}{n} - \frac{S_n^2}{n^2}$ with $T_n = \sum_i(-1)^{i+1}\varepsilon_i$, and $\mathrm{E}[(D_n - 1)^2] = \frac4n + \frac{3}{n^2} - \frac{2}{n^3}$. (e) [Both readings fail.] $\Pr(|D_n| > 1/2) \ge 1 - 4\mathrm{E}[(D_n - 1)^2] \to 1$ along even $n$, so the consistency reading fails; and $\Pr(|D_n| < 3/2) \ge 1 - 4\mathrm{E}[(D_n - 1)^2] \to 1$, so the display as printed fails at $\epsilon = 3/2$. (f) [Proposition 2 itself is not contradicted.] In this example $\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^{\mathcal O};Z) > 1/2) \to 0$.

: SAP.P011a.T11Statement

Theorem: SAP.P011a.T11

Statement hash: 6bc8fe05eeb7cc132043def495a9c244334db588451b0d8c5f06fd62ce1b0afd

Statement definition: SAP.P011a.T11Statement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:25:11Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T12

Proposition 2 with its proof corrected (a direct one-sided argument that replaces the auxiliary consistency step)

p. 2246 (PDF p. 9)

p. 2246 (PDF p. 9). Under Assumptions 2 and 3, the homoskedastic oracle estimator satisfies the entire finite moment, covariance and scalar gap chain, the explicit tail bound 4(1+c2)^3 c1^4/(n epsilon^2), and one-sided asymptotic conservativeness. Unit means may differ, early common variances may vanish, and neither the mean spread nor the common variance needs a limit.

Historical component credits

Proved AronowAudit.P011.homoskedastic_repair Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; No equal-unit-mean, positive early variance, auxiliary-estimator consistency or separate mean-spread limit is assumed. The intermediate chain inequalities in T12(ii),(iii) follow algebraically from the exported finite bounds and definitions.
Proved AronowAudit.P011.one_sided_algebra Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The statement matches the assigned finite auxiliary component; no hidden result assumption or problematic totalization is used.
Proved AronowAudit.P011.centered_sum_zero Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.
Proved AronowAudit.P011.sample_variance_centering Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its helper or component is already supplied by the complete component rows used here.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:07f73cdfbe863ce127d44df5e96ed92ee6534e67ffa994ad41723d2c79a058b5

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the validator's repair] Proposition 2 stands as printed: given Assumptions 2 and 3, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^{\mathcal O};Z) > \epsilon) = 0$ for any $\epsilon > 0$. Its proof is replaced by the following chain, in which $X_i = \theta_i Y_i$, $\delta_i = \mathrm{E}[X_i] - \beta$, $\sigma_n^2 = \mathrm{var}(X_i)$ is the common variance at sample size $n$ (Assumption 3 for $i, j \in \mathcal V_S$), $M_n = \frac1n\sum_i\delta_i^2 \ge 0$, $g_n = 1 + \frac1n\sum_i\sum_j A^{\mathcal O}_{ij}$, $U_n = \frac1n\sum_i(X_i - \beta)^2$ and $R_n = \hat\sigma_\theta^2 - (\sigma_n^2 + M_n)$. (i) $\hat\sigma_\theta^2 = U_n - (\hat\beta - \beta)^2$; $\mathrm{E}[U_n] = \sigma_n^2 + M_n$; $\mathrm{var}(U_n) \le \frac{(1 + c_2)c_1^4}{n}$; $\mathrm{E}[(\hat\beta - \beta)^2] \le \frac{(1 + c_2)c_1^2}{n}$; $\mathrm{E}[(\hat\beta - \beta)^4] \le \frac{(1 + c_2)c_1^4}{n}$; hence $\mathrm{E}[R_n^2] \le \frac{4(1 + c_2)c_1^4}{n}$. (ii) $n\,\mathrm{var}(\hat\beta) = \frac1n\sum_i\sum_j\mathrm{cov}(X_i, X_j) \le \sigma_n^2g_n$, and $1 \le g_n \le 1 + c_2$. (iii) $n\widehat V_2(A^{\mathcal O};Z) = g_n\hat\sigma_\theta^2$, and in every realization $n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^{\mathcal O};Z) \le g_n(\sigma_n^2 - \hat\sigma_\theta^2) = -g_n(M_n + R_n) \le (1 + c_2)|R_n|$. (iv) For every $\epsilon > 0$ and every $n$: $\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^{\mathcal O};Z) > \epsilon) \le \frac{4(1 + c_2)^3c_1^4}{n\epsilon^2}$. The proof uses no condition on the unit means, no limit for $M_n$ or $\sigma_n^2$, no auxiliary correlations and not Proposition 1. The consistency of $n\widehat V_2^*(A^{\mathcal O};Z)$ for $n\,\mathrm{var}(\hat\beta)$ asserted in the printed proof is not claimed: it is false in general when the unit means differ (P011-T11).

: SAP.P011a.T12WholeStatement

Theorem: SAP.P011a.T12

Statement hash: 4af61e7defc80fcd8a46fcb7297ed75ed0a736e4fcc842d88b3c8dd2f997e794

Statement definition: SAP.P011a.T12WholeStatement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:25:11Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.homoskedastic_repair: observed not used

AronowAudit.P011.one_sided_algebra: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T13

The complexity of programs (3.5), (3.6) and (3.8), corrected: exact solution by reduction to maximum-weight matching

p. 2245 (PDF p. 8), the two paragraphs after program (3.6); programs (3.5) p. 2244 (PDF p. 7), (3.6) p. 2245 (PDF p. 8), (3.8) p. 2246 (PDF p. 9)

p. 2245 (PDF p. 8), the two paragraphs after program (3.6); programs (3.5) p. 2244 (PDF p. 7), (3.6) p. 2245 (PDF p. 8), (3.8) p. 2246 (PDF p. 9). The finite degree-constrained optimization problem reduces to a matching gadget with exact decoding, counts, and two attained numerical instances.

Historical component credits

Proved AronowAudit.P011.Wave1.matching_reduction_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The feasible optional decomposition is existential rather than written with unique existence; forced/optional disjointness makes its witness uniquely S minus F. No time-complexity claim is assigned to this row.
Proved AronowAudit.P011.Wave1.corrected_polynomial_size_matching_reduction Withdrawn support; baseline fidelity: wrong_credit; manifest: not used; not used in Solution; Excluded: final_status is wrong_credit; no independent target credit is taken.
Proved AronowAudit.P011.matching_gadget_exact_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is only the exact finite-cardinality component; the tagged types really are finite, including the empty cases.
Proved AronowAudit.P011.Wave1.matching_exact_instances_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The existential matching witnesses certify attained optima and are accompanied by universal optimality; the result is not just an unattained upper bound.

Formal result: restricted

Scope: Retains the finite feasibility test, existential mandatory/optional decomposition, optimal decoding and objective recovery for every supplied maximum matching, exact gadget counts and combinatorial size bounds, and both exact attained instances. Drops (a)'s equivalence with the paper's programs, (b)'s explicit uniqueness and separate optional-optimum identity, a general existence assertion for maximum matchings and an independently defined optimum-value identity in (c), and all runtime and complexity-class conclusions of (e). No encoded-bit-length conclusion is stated.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:bdc64eb35c0b2a4172b01662eaef65eef1d402f16e7bcdd775d818e00aa662ad

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected text, from the validator] "This program can be encoded as a special multidimensional knapsack instance. Its degree-incidence structure makes it a maximum-weight simple b-matching problem. After fixing mandatory edges and subtracting their incidences from the degree bounds, it admits an exact polynomial-time solution by reduction to ordinary maximum-weight matching. Program (3.8) is its unit-weight case." [Formal content] Let $V = \{1, \ldots, n\}$, $K = \{\{i,j\} : 1 \le i < j \le n\}$, $F = \mathcal E_R \subseteq K$ the mandatory edges, $d_1, \ldots, d_n$ nonnegative integers, and $w_e$ ($e \in K$) rational weights. Problem (P): maximize $\sum_{e \in S}w_e$ over $S \subseteq K$ subject to $F \subseteq S$ and $\deg_S(i) \le d_i$ for every $i \in V$. (a) [(P) is the paper's program.] For a binary symmetric zero-diagonal $A$ with edge set $S$ and $r_i = \theta_i Y_i - \hat\beta$: $r^{\mathsf T}Ar = 2\sum_{\{i,j\} \in S}r_ir_j$ and $1^{\mathsf T}A1 = 2|S|$. So with $w_{\{i,j\}} = r_ir_j$ the maximizers of (P) are exactly the maximizers of (3.5), equivalently of (3.6) with $\hat v_{ij} = r_ir_j$; with all $w_e = 1$ they are the maximizers of (3.8). (b) [Mandatory edges.] Let $\bar d_i = \min\{d_i, n-1\}$, $f_i = \deg_F(i)$, $b_i = \bar d_i - f_i$ and $E = K \setminus F$. (P) is feasible if and only if $b_i \ge 0$ for all $i$. Every feasible $S$ is uniquely $F \cup T$ with $T \subseteq E$ and $\deg_T(i) \le b_i$, and $\mathrm{OPT}(\mathrm P) = C_F + \mathrm{OPT}_b$, where $C_F = \sum_{e \in F}w_e$ and $\mathrm{OPT}_b = \max\{\sum_{e \in T}w_e : T \subseteq E,\ \deg_T(i) \le b_i \text{ for all } i\}$. (c) [The matching instance.] Assume (P) feasible. Let $q = |E|$, $L$ the least common multiple of the denominators of the $w_e$, $e \in E$ ($L = 1$ if $E = \emptyset$), $W_e = Lw_e \in \mathbb Z$ and $M = 1 + \max_{e \in E}|W_e|$ (the maximum of the empty set being 0). Let $H$ be the graph with $b_i$ slot vertices $s_{i,1}, \ldots, s_{i,b_i}$ for each $i \in V$; two gadget vertices $p_{e,i}, p_{e,j}$ for each $e = \{i,j\} \in E$; an internal edge $p_{e,i}p_{e,j}$ of weight $4M$; external edges from $p_{e,i}$ to every slot of $i$ and from $p_{e,j}$ to every slot of $j$, each of weight $2M + W_e$; and no other edges. Then $\mathrm{OPT}(\mathrm P) = C_F + \frac{\mathrm{OPT}_{\mathrm{match}}(H) - 4Mq}{2L}$, where $\mathrm{OPT}_{\mathrm{match}}(H)$ is the maximum total weight of a matching of $H$; and for every maximum-weight matching $J$ of $H$, the set $F \cup T_J$, with $T_J$ the set of $e \in E$ both of whose gadget vertices are matched to slots, is an optimal solution of (P). (d) [Size.] $|V(H)| = 2q + \sum_i b_i \le 2n(n-1)$ and $|E(H)| = q + \sum_{\{i,j\} \in E}(b_i + b_j) \le q(2n - 1)$. (e) [Consequence.] Programs (3.5), (3.6) and (3.8) are solved exactly in time polynomial in the input size by any polynomial-time maximum-weight matching algorithm; if their decision versions were NP-hard, then P = NP. (f) [Exact instances from the validator.] $n = 4$, $r = (3/2, 1/2, -1/2, -3/2)$, $w_{ij} = r_ir_j$ (so $w_{12} = w_{34} = 3/4$, $w_{13} = w_{24} = -3/4$, $w_{14} = -9/4$, $w_{23} = -1/4$). With $d = (2, 1, 2, 7)$ and $F = \{\{1,3\}, \{3,4\}\}$: $b = (1, 1, 0, 2)$, $L = 4$, $M = 10$, $q = 4$, $\mathrm{OPT}_{\mathrm{match}}(H) = 166$, $4Mq = 160$, $\mathrm{OPT}(\mathrm P) = 3/4$, attained by $\{\{1,2\}, \{1,3\}, \{3,4\}\}$. With $d = (1, 1, 1, 1)$ and $F = \{\{1,3\}\}$: $b = (0, 1, 0, 1)$, $q = 5$, $\mathrm{OPT}_{\mathrm{match}}(H) = 4Mq = 200$, $\mathrm{OPT}(\mathrm P) = -3/4$, attained by $F$ alone.

: SAP.P011a.T13Statement

Theorem: SAP.P011a.T13

Statement hash: 54a0ef9ea4a46d39825ab4691dd5d3677eccf33a396a09e73aed1962c218beb0

Statement definition: SAP.P011a.T13Statement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:25:11Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T14

The linear-programming relaxation of programs (3.5), (3.6) and (3.8), corrected description: an upper bound that need not be attained by a graph

p. 2245 (PDF p. 8), after program (3.6); p. 2248 (PDF p. 11), "Upper bounds for the variance estimates can be obtained by solving a relaxed form of the programs (3.5) and (3.8)"; Appendix B, p. 2250 (PDF p. 13)

p. 2245 (PDF p. 8), after program (3.6); p. 2248 (PDF p. 11), "Upper bounds for the variance estimates can be obtained by solving a relaxed form of the programs (3.5) and (3.8)"; Appendix B, p. 2250 (PDF p. 13). Certified relaxed bounds control integer objectives and both variance estimators, and the six-unit and triangle examples have strictly fractional optima.

Historical component credits

Proved AronowAudit.P011.relaxation_upper_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The universal relaxed-bound premise is the certified input specified by the target; it is not the desired integer bound assumed as a hypothesis. Compatibility is the actual graph feasibility condition.
Proved AronowAudit.P011.relaxation_gaps Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; These are witnesses for the corrected nonintegrality claim, with upper bounds and attainment both exported. The three-unit unit-weight relaxed objective 3/2 is the ordered program value 3.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:57fb0118f3f1c6e5e9a82822d705a08c083f794eac9d8f5284dcb77b239f9430

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected text, from the validator] "General integer-programming solvers remain an implementation option; the degree-only LP relaxation can have fractional optima and still provides an upper bound when optimized or certified." [Formal content] The relaxation of (3.6) replaces $a_{ij} \in \{0, 1\}$ by $0 \le a_{ij} \le 1$, keeping the degree constraints and $a_{ij} = 1$ for $\{i,j\} \in \mathcal E_R$. (a) [Upper bound.] The optimal value $w^{\mathrm{LP}}$ of the relaxation is at least the optimal value $w^{\mathrm{IP}}$ of (3.6). Hence $\frac{1}{n^2}\left[n\hat\sigma_\theta^2 + 2w^{\mathrm{LP}}\right] \ge \widehat V_1(A^m;Z)$ with $A^m \in \mathcal A_1^m$ when $\hat v_{ij} = (\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)$, and $\frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{2w^{\mathrm{LP}}}{n}\right] \ge \widehat V_2(A^m;Z)$ with $A^m \in \mathcal A_2^m$ when $\hat v_{ij} = 1$. The same holds with $w^{\mathrm{LP}}$ replaced by any certified upper bound on it; the objective value of an arbitrary feasible point of the relaxation is not such a bound. (b) [The relaxation is not integral for the objective of (3.5).] Six observed units, $\theta_i = 1$, $Y = (1, 1, 1, -1, -1, -1)$, $\hat\beta = 0$, every $d_i = 1$, $\mathcal E_R = \emptyset$. The weights $\hat v_{ij} = Y_iY_j$ are $1$ on the six within-sign pairs and $-1$ on the nine cross-sign pairs. $w^{\mathrm{IP}} = 2$ (for example the edges $\{1,2\}$ and $\{4,5\}$), so the optimum of (3.5) is 4. $w^{\mathrm{LP}} = 3$, attained by $a_{ij} = 1/2$ on the six within-sign pairs and 0 elsewhere, and bounded above by $\sum_e\hat v_ea_e \le \sum_e a_e = \frac12\sum_i\deg_a(i) \le \frac12\sum_i d_i = 3$; the relaxed optimum of (3.5) is 6. Formula (3.1) gives $\widehat V_1 = (6 + 4)/36 = 5/18$ at the integer optimum and $(6 + 6)/36 = 1/3$ at the relaxed optimum. (c) [The same for (3.8).] Three units with $d = (1, 1, 1)$ and $\mathcal E_R = \emptyset$: the optimum of (3.8) is $1^{\mathsf T}A1 = 2$ (one edge), and the optimum of its relaxation is 3 ($a_{12} = a_{13} = a_{23} = 1/2$; adding the three degree constraints gives $2\sum_e a_e \le 3$).

: SAP.P011a.T14Statement

Theorem: SAP.P011a.T14

Statement hash: d438da1956f73ccab4a0e96f58665c267a10fdbd62ae7f41b854e17edcb601fa

Statement definition: SAP.P011a.T14Statement

Certificate. 20261104T173846Z

Declaration review: faithful; 2026-11-04T18:25:11Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T15

The claim that the general estimator, applied with raw outcomes to an OLS coefficient under forced recruitment edges, is conservative (package README and vignette; Corollary 1 for the regression example of Section 2)

Package README.md, line 3, and vignettes/replication.Rmd, lines 100–105 (staged/packages/depinf/jrzubizarreta-depinf-f11933a); the results invoked are Corollary 1, p. 2245 (PDF p. 8), and the regression example, pp. 2240–2241 (PDF pp. 3–4)

package README line 3 and vignette lines 100–105; Section 2, pp. 2240–2241; Corollary 1, p. 2245. The two correctly specified binary star regressions satisfy the standing assumptions, full-rank design identities, independent Bernoulli leaf and conditional centre laws, and saturated binary and fractional constraints. Their raw maximal variance estimates converge below and above the true variance, respectively. The first has ratio 8/9, underestimation probability tending to one, and the stated strict undercoverage for both the clipped and printed-domain intervals; negative estimates vanish in probability. The explicit mean offsets have the stated negative and positive quadratics, while true-error products converge to the true variance in both models.

Historical component credits

Proved AronowAudit.P011.Wave1.star_counterexample_proof Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': "GLP011B4.lean:696 states the star-law counterexample, its binary-compatible-graph singleton, scaled variance limits and undercoverage, but contains no regression design, MeanModel assertion or OLS-intercept identity. Consequently it does not establish the correctly specified regressions in T15(a),(e) that the row's promised T15(a)-(f) package includes. The binary singleton at :707 also does not state T15(b)'s singleton for the real-valued LP relaxation. Those missing connections occur separately in P011_BlockOLSCompletion.lean:245 and P011_StarCompletion.lean:96; zero-based indexing and clipped coverage introduce no material difference here.", 'suggested_fix': 'Add conjuncts for both displayed two-column star designs, their Gram and mean-model identities, and their OLS intercepts equalling the sample mean. Also assert that every feasible real LP edge-weight vector equals the true adjacency vector, for both star families; the existing separate OLS and relaxation results supply these connections.', 'second_review': 'T15(a),(e) specify correctly specified two-column regressions whose OLS intercept is the sample mean, and T15(b) also requires the LP relaxation to have a single feasible point. P011/Wave1/Gap/GLP011B4.lean:696 exports the outcome-law counterexample and its limits, but its singleton at :707 concerns only Boolean graphs, and starModel at P011/P011_Audit.lean:424 contains no regression design or OLS assertion. The missing regression identities and real-valued relaxation singleton are substantive links to the credited regression counterexample, separately stated at P011/P011_BlockOLSCompletion.lean:245 and P011/P011_StarCompletion.lean:96. Conjoining those links repairs the statement without changing the counterexample.', 'severity': 'major'}
Proved AronowAudit.P011.star_mean_terms Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The signs are exact block calculations; their stochastic meaning uses the explicitly constructed mean vectors in the companion rows.
Proved AronowAudit.P011.block_ols_representations Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The path clauses are additional to the star component credited here. Positive K and the specified nonzero centered covariates exclude singular-inverse fallback.
Proved AronowAudit.P011.star_relaxation_singleton Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The row concerns only LP constraints, and does not claim an estimator or stochastic conclusion.
Proved AronowAudit.P011.saturated_graph_singleton Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the general finite constraint lemma used by the star witness, not a standalone stochastic counterexample.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f29af2e2ef7df43a36bbf7ed434bce5dd2d639f27b4a07cc430b04ed64bc204e

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Statement refuted] README.md, line 3: "The primary function, depvar, produces conservative variance estimates for generic linear unbiased estimators". The vignette applies it to regression coefficients with the recruitment edges forced: "# V1, some known dependencies" followed by depvar(Y, Theta[1, ]*n, d, GR, case = "heteroskedastic", ...). In the paper's terms this is Corollary 1 (p. 2245: "Given Assumption 2, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) = 0$ for any $\epsilon > 0$") for an OLS coefficient $\hat\beta_j = \frac1n\sum_i\Theta_{ji}Y_i$ of Section 2. [Counterexample, from the validator: binary outcomes, a correctly specified regression, forced recruitment stars] For $k \ge 1$ let $n = N_n = 4k$, in $k$ independent, identically distributed blocks. Each block has a centre (position 1) and three leaves (positions 2, 3, 4); the dependency graph and the recruitment graph are both the star with adjacency matrix $A_\ast = \begin{pmatrix}0&1&1&1\\1&0&0&0\\1&0&0&0\\1&0&0&0\end{pmatrix}$, and $d = (3, 1, 1, 1)$. The regressors are an intercept and $x = (3, -1, -1, -1)$ in each block, with true coefficients $(3/8, 1/8)$, so $\mathrm{E}[Y] = (3/4, 1/4, 1/4, 1/4)$ in each block. The leaves $L_1, L_2, L_3$ are independent Bernoulli$(1/4)$; given the leaves, the centre is Bernoulli with probability $p_C = \frac34 + \frac{1}{12}\sum_{j=1}^{3}(L_j - \frac14) \in [\frac{11}{16}, \frac{15}{16}]$. (a) [The hypotheses hold.] All outcomes are binary. Leaves are mutually independent and blocks are independent, so any two vertex sets with no star edge between them are independent (Definition 1). The block covariance matrix is $\Sigma_\ast = \begin{pmatrix}3/16&1/64&1/64&1/64\\1/64&3/16&0&0\\1/64&0&3/16&0\\1/64&0&0&3/16\end{pmatrix}$. Because $\sum_i x_i = 0$, $X^{\mathsf T}X = \mathrm{diag}(n, 3n)$ and the OLS intercept is the sample mean: $\theta_i = 1$, $\beta = 3/8$, $\mathrm{E}[\hat\beta] = \beta$. $n\,\mathrm{var}(\hat\beta) = \frac14 1^{\mathsf T}\Sigma_\ast 1 = \frac{27}{128}$. Assumption 2 holds with $c_1 = 1$, $c_2 = 3$, $c_3 = 27/128$, $c_4 = 2$. (b) [One compatible graph.] Every star edge is in $\mathcal E_R$ and every degree allowance is used by a forced edge, so $\mathcal A(Z) = \{A^{\mathcal O}\}$; the same holds for the linear-programming relaxation. (c) [The mean term.] $h := \mathrm{E}[Y] - \beta 1 = (3/8, -1/8, -1/8, -1/8)$ in each block and $h^{\mathsf T}(I + A_\ast)h = -3/32$, i.e. $-3/128$ per observation. (d) [Exact expectation and limit.] $\mathrm{E}[n\widehat V_1(A^{\mathcal O};Z)] = \frac{3(64k - 47)}{1024k}$ for every $k$, and $n\widehat V_1(A^m;Z) = n\widehat V_1(A^{\mathcal O};Z) \to \frac{27}{128} - \frac{3}{128} = \frac{3}{16}$ in probability, so the ratio of the limit to the true scaled variance is $8/9$. For $\epsilon = 3/256$, $\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) \to 1$. The limiting coverage of the interval of Corollary 3 is $2\Phi(z_{1-\alpha/2}\sqrt{8/9}) - 1 < 1 - \alpha$. (e) [The error can go either way.] With $x = (0, 1, -1, 0)$, coefficients $(1/2, 1/4)$, block means $(1/2, 3/4, 1/4, 1/2)$, leaves independent Bernoulli with means $3/4, 1/4, 1/2$ and centre probability $\frac12 + \frac{1}{12}\sum_j(L_j - \mathrm{E}[L_j]) \in [\frac38, \frac58]$, on the same forced stars: $h = (0, 1/4, -1/4, 0)$, $n\,\mathrm{var}(\hat\beta) = 47/192$ and $n\widehat V_1(A^m;Z) \to 53/192$ in probability. (f) [True-error products.] In both models, with $e_i^\ast = \theta_i(Y_i - x_i^{\mathsf T}b)$ ($b$ the true coefficient vector), $\frac1n e^{\ast\mathsf T}(I + A^{\mathcal O})e^\ast$ converges in probability to the true value ($27/128$, respectively $47/192$).

: SAP.P011b.T15WholeStatement

Theorem: SAP.P011b.T15

Statement hash: af58bb848664f18259442d3d74499b84fe12872a561f2de72651b8c662ec99b6

Statement definition: SAP.P011b.T15WholeStatement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T19:07:09Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T16

What can be claimed for an OLS coefficient: the condition for the raw-outcome estimator, and the residual-based estimator with the design conditions it needs

Regression example, pp. 2240–2241 (PDF pp. 3–4); (3.1), p. 2242 (PDF p. 5); package vignette vignettes/replication.Rmd, lines 100–105

regression example, pp. 2240–2241; (3.1), p. 2242; package vignette lines 100–105. Corrected claim. For a full-rank fixed-design OLS coefficient, weighted unit means and their offsets have the exact stated formulas. Under Assumption 2 the raw oracle has the moving mean-bias limit, is consistent exactly when that bias tends to zero, and is conservative exactly when its liminf is nonnegative. With valid constraints this condition suffices for every compatible maximum, but is not necessary in general, as the independent binary-star regression with empty recruitment demonstrates. Equal weighted unit means suffice for consistency and maximal conservativeness. Every design-space outcome shift leaves fitted residual products and all residual objectives unchanged and has the stated effect on raw centred products. With an intercept, constant shifts preserve slopes and residual objectives, but can change raw slope variance, as an explicit two-point example shows. Under Assumption 2, fixed dimension, uniformly bounded rows and outcomes, and a uniformly coercive normalized Gram matrix, OLS has the root-n rate. The displayed normalized residual perturbation inequality holds and its right side tends to zero in probability; the residual oracle is consistent and every compatible residual maximum is conservative. Deterministic coefficients and targets may vary with sample size. The usual measure-theoretic conditions are assumed.

Historical component credits

Proved AronowAudit.P011.Wave1.ols_repair_proof Limited support; baseline fidelity: narrower; manifest: not used; not used in Solution; {'source': 'adjudication+second review', 'adjudicator': "GLP011B5.lean:356 makes Scaling a premise of the entire conjunction, including the finite mean formula at :361 and exact shift identities at :364. T16(i)'s finite mean identity and T16(ii)'s shift identities require the full-rank integrable mean model, not Assumption 2; Gaussian-error regressions and deterministic regressions provide substantive excluded cases. The asymptotic clause at :369 exports the root-n rate, residual-oracle consistency and maximal conservativeness, but omits T16(iii)'s displayed normalized quadratic perturbation inequality and the convergence of its right-hand side. The separately stated finite identities and perturbation lemmas do not restore these conclusions to this credited declaration.", 'suggested_fix': 'State the exact mean and shift identities for a finite positive sample size under full rank, MeanModel and the OLS weight identity alone; require Scaling only for the asymptotic clauses. Under the additional bounds of T16(iii), also export the displayed inequality with factor 1+c2 and convergence in probability of its right-hand side to zero.', 'second_review': "T16(i)'s exact mean formula and T16(ii)'s shift identities require the full-rank integrable mean model, whereas P011/Wave1/Gap/GLP011B5.lean:356 assumes Scaling for the entire conjunction. Scaling at P011/P011_Audit.lean:187 adds uniformly bounded weighted outcomes and positive limiting scaled variance, excluding, for example, Gaussian-error and deterministic full-rank regressions from these finite identities. The asymptotic conclusion at GLP011B5.lean:369 also omits T16(iii)'s displayed normalized perturbation inequality and convergence of its right-hand side. Separate the finite identities from Scaling and export the perturbation conclusion under the stated asymptotic design bounds.", 'severity': 'major'}

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: source-claim-import-2026-11-08:P011-T16:1

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected claim, from the validator's repair] Let $X$ be a fixed $n \times p$ design with $X^{\mathsf T}X$ invertible, $\mathrm{E}[Y] = Xb$, $\hat b = (X^{\mathsf T}X)^{-1}X^{\mathsf T}Y$, $\Theta = n(X^{\mathsf T}X)^{-1}X^{\mathsf T}$, and fix a coefficient $j$: $\theta_i = \Theta_{ji}$, $\hat\beta = \hat b_j = \frac1n\sum_i\theta_iY_i$, $\beta = b_j$. (i) [The raw-outcome estimator.] $\mathrm{E}[\theta_iY_i] = \Theta_{ji}x_i^{\mathsf T}b$, so $\delta_i = \Theta_{ji}x_i^{\mathsf T}b - b_j$, which in general is not 0. By P011-T3, under Assumption 2 the estimator (3.1) with raw outcomes satisfies $n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) - \frac1n\delta^{\mathsf T}(I + A^{\mathcal O})\delta \to 0$ in probability; the conclusion of Proposition 1 for an OLS coefficient holds if and only if $\frac1n\delta^{\mathsf T}(I + A^{\mathcal O})\delta \to 0$. The raw oracle is asymptotically conservative if and only if $\liminf_{n\to\infty}\frac1n\delta^{\mathsf T}(I + A^{\mathcal O})\delta \ge 0$; under valid constraints, this latter condition is sufficient for the conclusion of Corollary 1 for every raw compatible maximizer, but is not necessary in general. The transparent sufficient condition for the conclusion of Proposition 1 and, under valid constraints, for the conclusion of Corollary 1 for every raw compatible maximizer is $\mathrm{E}[\theta_iY_i] = \beta$ for all units and every $n$. (ii) [Residual products; exact.] Let $q_i = \theta_i(Y_i - x_i^{\mathsf T}\hat b)$ and $\widehat V_1^{\mathrm{res}}(A;Z) = \frac{1}{n^2}q^{\mathsf T}(I + A)q$. Replacing $Y$ by $Y + Xc$ for any $c \in \mathbb R^p$ leaves $q$, and hence $\widehat V_1^{\mathrm{res}}(A;Z)$ for every $A$, unchanged, whereas the raw centred products $\theta_iY_i - \hat\beta$ change by $\theta_ix_i^{\mathsf T}c - c_j$. In particular, when the design contains an intercept, adding a constant to every outcome changes neither the slope estimates nor $\widehat V_1^{\mathrm{res}}$, but can change $\widehat V_1(A;Z)$ of (3.1) for a slope coefficient. (iii) [Residual oracle; asymptotic.] Suppose Assumption 2 holds for this coefficient and, in addition, the dimension $p$ is fixed, the rows $x_i$ and the outcomes $Y_i$ are bounded uniformly in $i$ and $n$, and the smallest eigenvalue of $X^{\mathsf T}X/n$ is bounded away from zero. Then $\hat b - b = O_p(n^{-1/2})$; with $e_i^\ast = \theta_i(Y_i - x_i^{\mathsf T}b)$ and $\Delta_i = \theta_ix_i^{\mathsf T}(\hat b - b)$, $\frac1n\left|(e^\ast - \Delta)^{\mathsf T}(I + A^{\mathcal O})(e^\ast - \Delta) - e^{\ast\mathsf T}(I + A^{\mathcal O})e^\ast\right| \le (1 + c_2)\left[2\frac{\|e^\ast\|}{\sqrt n}\frac{\|\Delta\|}{\sqrt n} + \frac{\|\Delta\|^2}{n}\right] \to 0$ in probability; hence $n\widehat V_1^{\mathrm{res}}(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) \to 0$ in probability, and for a maximizer $A^{m,\mathrm{res}}$ of $\widehat V_1^{\mathrm{res}}(A;Z)$ over $\mathcal A(Z)$ (valid constraints), $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1^{\mathrm{res}}(A^{m,\mathrm{res}};Z) > \epsilon) = 0$ for any $\epsilon > 0$.

: SAP.P011b.T16WholeStatement_W2

Theorem: SAP.P011b.T16

Statement hash: 313d155345a1f1c79e1375d3a8b32ab24113b4ee5c1eac984b227b78803de64e

Statement definition: SAP.P011b.T16WholeStatement_W2

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T19:34:51Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.Wave1.ols_repair_proof: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T17

Display (3.3), corrected, with the leading term of (3.4)

(3.3): p. 2243 (PDF p. 6); (3.4): pp. 2243–2244 (PDF pp. 6–7)

(3.3), p. 2243; (3.4), pp. 2243–2244. The corrected pair and fourth moments and constant-centering covariance expansion hold on their separate integrable moment domains.

Historical component credits

Proved AronowAudit.P011.Wave1.corrected_covariance_expansion_moments Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': 'P011/Wave1/Gap/Fid2P011.lean:423 gives independently conditional weighted-pair, weighted-fourth, constant-centering, raw-pair and raw-fourth identities. These antecedents require the individual moments displayed in the respective T17 formulas, rather than global raw L4, and the random-centering branches explicitly identify covOf with ordinary covariance. Grouping the sixteen terms and the two pair blocks changes no coefficients or cases; the constant-centering moment domain likewise justifies ordinary covariance, and the asymptotic remainder is outside this target.'}
Proved AronowAudit.P011.fourth_polynomial Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The complete moment-domain result already supplies the fourth-moment identity; this polynomial row is not needed separately.
Proved AronowAudit.P011.covariance_expansion_linear Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The complete moment-domain result already supplies the covariance expansion; this abstract algebra row is not needed separately.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:04f57efa7aa174f8147649aa211992e68ace88372a44be4b4dfcba2d673c7a0a

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected statement, from the validator] With $X_i = \theta_iY_i$, for any indices $i, j, k, l \in \mathcal V_S$: (i) $\xi_{ijkl} = \mathrm{cov}\big((X_i - \hat\beta)(X_j - \hat\beta), (X_k - \hat\beta)(X_l - \hat\beta)\big) = \mathrm{E}\big[(X_i - \hat\beta)(X_j - \hat\beta)(X_k - \hat\beta)(X_l - \hat\beta)\big] - Q_{ij}Q_{kl}$, where $Q_{ij} = \mathrm{E}[(X_i - \hat\beta)(X_j - \hat\beta)] = \theta_i\theta_j\mathrm{E}[Y_iY_j] - \theta_i\mathrm{E}[Y_i\hat\beta] - \theta_j\mathrm{E}[Y_j\hat\beta] + \mathrm{E}[\hat\beta^2]$. (ii) Written out, the corrected display (3.3) is: $\xi_{ijkl} = \mathrm{E}\big[(\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)(\theta_kY_k - \hat\beta)(\theta_lY_l - \hat\beta)\big] - \mathrm{E}\big[(\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)\big]\mathrm{E}\big[(\theta_kY_k - \hat\beta)(\theta_lY_l - \hat\beta)\big]$ $= \theta_i\theta_j\theta_k\theta_l\mathrm{E}[Y_iY_jY_kY_l] - \theta_i\theta_j\theta_k\mathrm{E}[Y_iY_jY_k\hat\beta] - \theta_i\theta_j\theta_l\mathrm{E}[Y_iY_jY_l\hat\beta] - \theta_i\theta_k\theta_l\mathrm{E}[Y_iY_kY_l\hat\beta] - \theta_j\theta_k\theta_l\mathrm{E}[Y_jY_kY_l\hat\beta] + \theta_i\theta_j\mathrm{E}[Y_iY_j\hat\beta^2] + \theta_i\theta_k\mathrm{E}[Y_iY_k\hat\beta^2] + \theta_i\theta_l\mathrm{E}[Y_iY_l\hat\beta^2] + \theta_j\theta_k\mathrm{E}[Y_jY_k\hat\beta^2] + \theta_j\theta_l\mathrm{E}[Y_jY_l\hat\beta^2] + \theta_k\theta_l\mathrm{E}[Y_kY_l\hat\beta^2] - \theta_i\mathrm{E}[Y_i\hat\beta^3] - \theta_j\mathrm{E}[Y_j\hat\beta^3] - \theta_k\mathrm{E}[Y_k\hat\beta^3] - \theta_l\mathrm{E}[Y_l\hat\beta^3] + \mathrm{E}[\hat\beta^4] - \big[\theta_i\theta_j\theta_k\theta_l\mathrm{E}[Y_iY_j]\mathrm{E}[Y_kY_l] - \theta_i\theta_j\theta_k\mathrm{E}[Y_iY_j]\mathrm{E}[Y_k\hat\beta] - \theta_i\theta_j\theta_l\mathrm{E}[Y_iY_j]\mathrm{E}[Y_l\hat\beta] + \theta_i\theta_j\mathrm{E}[Y_iY_j]\mathrm{E}[\hat\beta^2] - \theta_i\theta_k\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_kY_l] + \theta_i\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_l\hat\beta] + \theta_i\theta_k\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_k\hat\beta] - \theta_i\mathrm{E}[Y_i\hat\beta]\mathrm{E}[\hat\beta^2] - \theta_j\theta_k\theta_l\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_kY_l] + \theta_j\theta_l\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_l\hat\beta] + \theta_j\theta_k\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_k\hat\beta] - \theta_j\mathrm{E}[Y_j\hat\beta]\mathrm{E}[\hat\beta^2] + \theta_k\theta_l\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_kY_l] - \theta_k\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_k\hat\beta] - \theta_l\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_l\hat\beta] + \mathrm{E}[\hat\beta^2]\mathrm{E}[\hat\beta^2]\big]$. The three changes from the print are: $\theta_j$ restored before $Y_j$ in the first expectation; $\theta_i$ restored in the term $-\theta_i\theta_k\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_kY_l]$; and the opening bracket moved to stand immediately after the minus sign, before $\theta_i\theta_j\theta_k\theta_l$. (iii) [The leading term of (3.4); exact for a constant $\beta$.] $\mathrm{cov}\big((X_i - \beta)(X_j - \beta), (X_k - \beta)(X_l - \beta)\big) = \mathrm{cov}(X_iX_j, X_kX_l) - \beta\big[\mathrm{cov}(X_iX_j, X_k) + \mathrm{cov}(X_iX_j, X_l) + \mathrm{cov}(X_i, X_kX_l) + \mathrm{cov}(X_j, X_kX_l)\big] + \beta^2\big[\mathrm{cov}(X_i, X_k) + \mathrm{cov}(X_i, X_l) + \mathrm{cov}(X_j, X_k) + \mathrm{cov}(X_j, X_l)\big]$, which is the expression $\xi'_{ijkl}$ printed in (3.4).

{n : ℕ} (hn : 0 < n) (m : Model n) (i j k l : Fin n) : SAP.P011b.T17Statement hn m i j k l

Theorem: SAP.P011b.T17

Statement hash: c2ad8aa50cc9912cdcdf57fcd2f1b96826d9a77dad7093b6c5513a81d97a0214

Statement definition: SAP.P011b.T17Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T19:07:09Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.fourth_polynomial: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T18

Display (3.3), as printed

p. 2243 (PDF p. 6)

The printed claim is false: The constant-outcome witness gives true covariance zero, printed values 9/4 and 93009/16, the named bracket-only repair -70, and fully corrected value zero. (3.3), p. 2243.

Historical component credits

Counterexample proved AronowAudit.P011.printed_expansion_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P011/P011_Audit.lean:1260 gives the target's probability-one constant witness and the exact values 0, 9/4, 93009/16, -70 and 0. The bracket-only repair is represented by precisely the evaluated rational expression displayed in T18(d), while the other printed and corrected expressions retain their moment definitions. A Dirac probability law is a legitimate witness to the finite algebraic failure; the constant witness does not rely on a zero-measure convention."}

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:a734ef75231bfb7a20ec2ebbf5fabafe62b82dee851d238f3e7a306a1abf8c65

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed statement refuted, p. 2243] $\xi_{ijkl} = \mathrm{cov}\big((\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta), (\theta_kY_k - \hat\beta)(\theta_lY_l - \hat\beta)\big)$ $= \mathrm{E}\big[(\theta_iY_i - \hat\beta)(Y_j - \hat\beta)(\theta_kY_k - \hat\beta)(\theta_lY_l - \hat\beta)\big] - \mathrm{E}\big[(\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)\big]\mathrm{E}\big[(\theta_kY_k - \hat\beta)(\theta_lY_l - \hat\beta)\big]$ $= \theta_i\theta_j\theta_k\theta_l\mathrm{E}[Y_iY_jY_kY_l] - \theta_i\theta_j\theta_k\mathrm{E}[Y_iY_jY_k\hat\beta] - \theta_i\theta_j\theta_l\mathrm{E}[Y_iY_jY_l\hat\beta] - \theta_i\theta_k\theta_l\mathrm{E}[Y_iY_kY_l\hat\beta] - \theta_j\theta_k\theta_l\mathrm{E}[Y_jY_kY_l\hat\beta] + \theta_i\theta_j\mathrm{E}[Y_iY_j\hat\beta^2] + \theta_i\theta_k\mathrm{E}[Y_iY_k\hat\beta^2] + \theta_i\theta_l\mathrm{E}[Y_iY_l\hat\beta^2] + \theta_j\theta_k\mathrm{E}[Y_jY_k\hat\beta^2] + \theta_j\theta_l\mathrm{E}[Y_jY_l\hat\beta^2] + \theta_k\theta_l\mathrm{E}[Y_kY_l\hat\beta^2] - \theta_i\mathrm{E}[Y_i\hat\beta^3] - \theta_j\mathrm{E}[Y_j\hat\beta^3] - \theta_k\mathrm{E}[Y_k\hat\beta^3] - \theta_l\mathrm{E}[Y_l\hat\beta^3] + \mathrm{E}[\hat\beta^4] - \theta_i\theta_j\theta_k\theta_l\big[\mathrm{E}[Y_iY_j]\mathrm{E}[Y_kY_l] - \theta_i\theta_j\theta_k\mathrm{E}[Y_iY_j]\mathrm{E}[Y_k\hat\beta] - \theta_i\theta_j\theta_l\mathrm{E}[Y_iY_j]\mathrm{E}[Y_l\hat\beta] + \theta_i\theta_j\mathrm{E}[Y_iY_j]\mathrm{E}[\hat\beta^2] - \theta_k\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_kY_l] + \theta_i\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_l\hat\beta] + \theta_i\theta_k\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_k\hat\beta] - \theta_i\mathrm{E}[Y_i\hat\beta]\mathrm{E}[\hat\beta^2] - \theta_j\theta_k\theta_l\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_kY_l] + \theta_j\theta_l\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_l\hat\beta] + \theta_j\theta_k\mathrm{E}[Y_j\hat\beta]\mathrm{E}[Y_k\hat\beta] - \theta_j\mathrm{E}[Y_j\hat\beta]\mathrm{E}[\hat\beta^2] + \theta_k\theta_l\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_kY_l] - \theta_k\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_k\hat\beta] - \theta_l\mathrm{E}[\hat\beta^2]\mathrm{E}[Y_l\hat\beta] + \mathrm{E}[\hat\beta^2]\mathrm{E}[\hat\beta^2]\big]$ (3.3) [Counterexample, from the validator, with the value of the last line recomputed here] Take $n = 4$, degenerate outcomes $Y_1 = Y_2 = Y_3 = Y_4 = 1$ (constants), $\theta = (2, 3, 4, 5)$ and $(i, j, k, l) = (1, 2, 3, 4)$. Then $\hat\beta = 7/2$ is a constant and every expectation in (3.3) is the value of its argument. (a) The left side is a covariance of constants: $\xi_{1234} = 0$. (b) The second line as printed, with $(Y_j - \hat\beta)$ in place of $(\theta_jY_j - \hat\beta)$, equals $(2 - \tfrac72)(1 - \tfrac72)(4 - \tfrac72)(5 - \tfrac72) - (2 - \tfrac72)(3 - \tfrac72)(4 - \tfrac72)(5 - \tfrac72) = \frac{45}{16} - \frac{9}{16} = \frac94 \ne 0$. (c) The final expansion exactly as printed (the factor $\theta_i\theta_j\theta_k\theta_l = 120$ multiplying the whole bracket, whose first term is $\mathrm{E}[Y_iY_j]\mathrm{E}[Y_kY_l] = 1$ and whose fifth term lacks $\theta_i$) equals $\frac{9}{16} - 120\cdot\left(-\frac{775}{16}\right) = \frac{93009}{16} \ne 0$. (d) With the opening bracket moved before $\theta_i\theta_j\theta_k\theta_l$ but $\theta_i$ still missing from $-\theta_k\theta_l\mathrm{E}[Y_i\hat\beta]\mathrm{E}[Y_kY_l]$, the expansion equals $\frac{9}{16} - \left(\frac{9}{16} + 140 - 70\right) = -70 \ne 0$. (e) With all three corrections of P011-T17 the expansion equals $\frac{9}{16} - \frac{9}{16} = 0$.

: SAP.P011b.T18Statement

Theorem: SAP.P011b.T18

Statement hash: 7e8b671166976fc362dffd024d28e25da5c86753283c7e7291e4820994510650

Statement definition: SAP.P011b.T18Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T19:07:09Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T19

Proof of Proposition 2, the convergence display, with the inequality corrected

p. 2246 (PDF p. 9)

proof of Proposition 2, p. 2246. For positive common variance under Assumptions 2 and 3, the auxiliary error is k times the sample-variance error, where k is the true scaled variance divided by the common variance and lies between zero and 1+c2. The sample-variance remainder after subtracting between-unit mean spread converges to zero in probability. Vanishing mean spread, in particular equal weighted unit means, suffices for auxiliary consistency. The explicit alternating-mean independent-sign family satisfies all these standing assumptions with positive common variance, yet its mean spread tends to one and its auxiliary error exceeds 1/2 with probability tending to one.

Historical component credits

Proved AronowAudit.P011.auxiliary_corrected_display Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; All three positive assertions of the repaired display are covered, including the explicitly stated positive common variance. This theorem gives the sufficient mean-spread condition; the separate observation that sign repair alone fails is not independently refuted by any T19 row in this chunk.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:743a6770ee64ba9714eaf5c74271ddcf22405eea321906ccfcb9972750c5a704

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected display, from the validator] "$\lim_{n\to\infty}\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) = 0$" (the print has $< \epsilon$). [Statement with its hypotheses] Under Assumptions 2 and 3, with $X_i = \theta_iY_i$, common variance $\sigma_n^2 = \mathrm{var}(X_i) > 0$, pairwise correlations $\rho_{ij} = \mathrm{cov}(X_i, X_j)/\sigma_n^2$ (P011-T21) and $\widehat V_2^*(A^{\mathcal O};Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij}\rho_{ij}\right]$: (i) [exact] $n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta) = k_n\left(\hat\sigma_\theta^2 - \sigma_n^2\right)$, where $k_n = 1 + \frac1n\sum_i\sum_j A^{\mathcal O}_{ij}\rho_{ij} = n\,\mathrm{var}(\hat\beta)/\sigma_n^2$; (ii) $\hat\sigma_\theta^2 - \sigma_n^2 - M_n \to 0$ in probability, where $M_n = \frac1n\sum_i(\mathrm{E}[X_i] - \beta)^2$ (P011-T12 (i)); (iii) hence the corrected display holds, for every $\epsilon > 0$, if $M_n \to 0$, in particular if $\mathrm{E}[\theta_iY_i] = \beta$ for every $i$; correcting the inequality sign alone does not make it hold under Assumptions 2 and 3 only (P011-T11, where $M_n \to 1$).

: SAP.P011b.T19WholeStatement

Theorem: SAP.P011b.T19

Statement hash: 35d1e7c18b6d951a111667ca6a6f0bedf59dbdd4349c1d64e8fb3b6ee9ee487d

Statement definition: SAP.P011b.T19WholeStatement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T19:07:09Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T20

Proof of Proposition 2, the convergence display as printed (with $< \epsilon$ )

p. 2246 (PDF p. 9)

The printed claim is false: Fair signs refute the printed limit-zero display and give its exact inside-event probabilities and limit one. proof of Proposition 2, p. 2246.

Historical component credits

Counterexample proved AronowAudit.P011.Wave1.printed_sign_counterexample_full_printed_negation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The negated contract keeps the printed strict-inside event and zero limit. Pairwise-correlation and observed-coordinate conventions lose no cases relevant to the all-observed edgeless witness; no conditional published theorem is assumed.
Proved AronowAudit.P011.auxiliary_empty_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The equality is samplewise and independent of moment assumptions, including if totalized covariance values are otherwise uninformative.
Proved AronowAudit.P011.uniform_coordinate_dependency Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The finite-uniform generality is exactly the auxiliary fact described by this row and specializes to fair signs; it does not claim the full asymptotic refutation.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:8e12272928fb7622cd5e819ad570b929f1dd13ca9954fef0b515b41084d52ae4

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed statement refuted, p. 2246] "$\lim_{n\to\infty}\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| < \epsilon) = 0$." [Counterexample, from the validator] Let $Y_1, Y_2, \ldots$ be independent with $\Pr(Y_i = 1) = \Pr(Y_i = -1) = 1/2$, $\theta_i = 1$, and for every $n$ the edgeless graph on $\{1, \ldots, n\}$ with every vertex observed, $\mathcal E_R = \emptyset$, $A^{\mathcal O} = 0$, $d_i = 0$. Then $\beta = 0$, $\mathrm{E}[\theta_iY_i] = \beta$ for every $i$, $|\theta_iY_i - \beta| = 1$, $\mathrm{var}(\theta_iY_i) = 1$ for every $i$, and $n\,\mathrm{var}(\hat\beta) = 1$: Assumptions 2 and 3 hold ($c_1 = 1$, $c_2 = 1$, $c_3 = 1$, $c_4 = 2$). (a) $n\widehat V_2^*(A^{\mathcal O};Z) = \hat\sigma_\theta^2 = 1 - \hat\beta^2$, so $|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| = \hat\beta^2$. (b) $\mathrm{E}[\hat\beta^2] = 1/n$, so for every $\epsilon > 0$: $\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| < \epsilon) \ge 1 - \frac{1}{n\epsilon} \to 1$, not 0. (c) Exact values at $\epsilon = 1/2$: the probability is $1/2$ at $n = 2$, $7/8$ at $n = 4$ and $32631/32768$ at $n = 16$.

: SAP.P011b.T20Statement

Theorem: SAP.P011b.T20

Statement hash: b9d342d6d8a236c169056c6acedc3737f50d84fd4169e73d246c9b06f80630f3

Statement definition: SAP.P011b.T20Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:54:55Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T21

Proof of Proposition 2, the correlations of the auxiliary estimator, with the pair index restored

p. 2246 (PDF p. 9)

proof of Proposition 2, p. 2246. Pairwise correlations give the exact variance identity and oracle bound, and the path witness has distinct adjacent correlations. The usual measure-theoretic conditions, including finite second moments of the weighted outcomes, are assumed.

Historical component credits

Proved AronowAudit.P011.pairwise_correlation_repair Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P011/P011_CorrelationCompletion.lean:132 states exactly T21(i)-(ii): the correlation-weighted variance identity, all pairwise correlations at most one, and the samplewise estimator inequality. Its L2 condition is on weighted coordinates and follows from the standing bounded-weighted-outcome assumption; it also expresses the finite variances needed by the target's covariance ratios. The definition at P011/P011_Audit.lean:1277 preserves the required zero-common-variance convention and adds no positive-variance or common-mean restriction."}
Proved AronowAudit.P011.path_correlation_instance Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Zero-based path indices correspond exactly to the printed 1-based indices; the two edge correlations differ, as required.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:2579781d01d073d0e3e5cae06bd551115e5e532a2ad7506b8877014471d4decd

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected text, from the validator] Define $X_i = \theta_iY_i$, $\sigma_\theta^2 = \mathrm{var}(X_i)$ (the common variance of Assumption 3), and $\rho_{ij} = \mathrm{cov}(X_i, X_j)/\sigma_\theta^2$ when $\sigma_\theta^2 > 0$ ($\rho_{ij} = 0$ by convention if $\sigma_\theta^2 = 0$); replace all four occurrences of $\rho_i$ in the proof by $\rho_{ij}$. The auxiliary estimator is $\widehat V_2^*(A^{\mathcal O};Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij}\rho_{ij}\right]$, and "all $\rho_{ij} \le 1$". [Formal content] Under Assumption 3 for $i, j \in \mathcal V_S$ and Definition 1: (i) $n\,\mathrm{var}(\hat\beta) = \sigma_\theta^2\left[1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij}\rho_{ij}\right]$; (ii) $\rho_{ij} \le 1$ for all $i, j$, and therefore $\widehat V_2^*(A^{\mathcal O};Z) \le \widehat V_2(A^{\mathcal O};Z)$ in every realization; (iii) [an instance showing that a single index cannot carry the correlations] on the path $1 - 2 - 3$ with $U, V$ independent, $\Pr(\cdot = 1) = \Pr(\cdot = -1) = 1/2$, and $X = (U, \tfrac35U + \tfrac45V, V)$: all three variances are 1, $\rho_{12} = 3/5$, $\rho_{23} = 4/5$, $\rho_{13} = 0$, and both sides of (i) equal $29/15$ ($n = 3$).

{n : ℕ} (hn : 0 < n) (m : SAP.P011b.Model n)
    (hdep : SAP.P011b.Dependency m) (hs : SAP.P011b.Homoskedastic m)
    (hL2 : ∀ i, MemLp (fun y => SAP.P011b.weighted m y i) 2 m.law) :
    SAP.P011b.T21Statement hn m hdep hs hL2

Theorem: SAP.P011b.T21

Statement hash: cfad696d2649255a5dbc20f5e07a25329ee40ef7e61046a41ea0ba4b5b4625a5

Statement definition: SAP.P011b.T21Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:54:55Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T22

The decision variables of program (3.6), corrected, and the equivalence of (3.5) and (3.6)

Sentence defining the decision variables: p. 2244 (PDF p. 7), last paragraph; program (3.6): p. 2245 (PDF p. 8)

decision-variable sentence, p. 2244; (3.6), p. 2245. Binary edge decisions and symmetric adjacency matrices describe the same finite programs, objectives and maximizers.

Historical component credits

Proved AronowAudit.P011.upper_program_equivalence Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The raw vector x stands for the arbitrary weighted observations theta_i Y_i. All positive sample sizes are allowed; n=0 is outside the statistical formula's domain.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f5dbb66a54704ec2c36a6d0a2982a9c02b13d42d0e298cee00b3c65693599edf

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected text, from the validator] "Define $\hat v_{ij} = (\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)$, and let $a_{ij} = 1$ if edge $\{i,j\}$ is selected in the candidate graph, and 0 otherwise, for $i < j$." [Formal content] Let $a = (a_{ij})_{1 \le i < j \le n} \in \{0, 1\}^{n(n-1)/2}$ and let $A$ be the $n \times n$ matrix with $A_{ij} = A_{ji} = a_{ij}$ for $i < j$ and $A_{ii} = 0$. The map $a \mapsto A$ is a bijection onto the binary symmetric zero-diagonal matrices, and under it: (i) $(\Theta Y - \hat\beta 1)^{\mathsf T}A(\Theta Y - \hat\beta 1) = 2\sum_{i=1}^{n}\sum_{j=i+1}^{n}\hat v_{ij}a_{ij}$; (ii) $(A1)_i = \sum_{j=1}^{i-1}a_{ji} + \sum_{j=i+1}^{n}a_{ij}$ for every $i$, so $A1 \preceq d$ (element-wise $\le$) is the degree constraint of (3.6); (iii) $A \succeq A_R$ if and only if $a_{ij} = 1$ for every $\{i,j\} \in \mathcal E_R$ with $i < j$; (iv) hence $A$ is optimal for (3.5) if and only if $a$ is optimal for (3.6) with the variables of the edges in $\mathcal E_R$ fixed at one; the optimal values differ by the factor 2; and $n^2\widehat V_1(A^m;Z) = n\hat\sigma_\theta^2 + 2w^\ast$ with $w^\ast$ the optimal value of (3.6); (v) with $\hat v_{ij} = 1$ for all $i < j$, $1^{\mathsf T}A1 = 2\sum_{i<j}a_{ij}$ and (3.6) is (3.8).

{n : ℕ} (hn : 0 < n) : SAP.P011b.T22Statement hn

Theorem: SAP.P011b.T22

Statement hash: a41bb79897bf672ae3ccc7bb0394f593df61e0547059daf2bcaf2e9a07d51ddf

Statement definition: SAP.P011b.T22Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:54:55Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T23

The decision variables of program (3.6), as printed

p. 2244 (PDF p. 7), last paragraph; program (3.6): p. 2245 (PDF p. 8)

decision-variable sentence, p. 2244; (3.6), p. 2245. Refutes interpreting the nonzero-product instruction as a data-determined feasible or optimal edge assignment: the specified three-vertex examples violate degree caps or omit a forced edge, and the unit-cap example has unique optimum edge23 with value 3/4.

Historical component credits

Counterexample proved AronowAudit.P011.printed_variables_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The object refuted is the finite program's proposed nonzero-product rule. No asymptotic or positive statistical theorem is inferred from this witness.
Proved AronowAudit.P011.printed_variables_optimum Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'P011/P011_Audit.lean:1348 states only that every compatible graph has objective at most 3/4 and that equality forces A = edge23. T23(c), also expressly claimed by this row, says that the optimum is 3/4 and is attained uniquely by edge23; the declaration does not assert feasibility of edge23 or attainment of 3/4. Those assertions are exported by the companion printed_variables_counterexample at line 1317, but they are absent from this credited declaration, so its standalone conclusion omits the existence/attainment part of the assigned claim.', 'suggested_fix': "Conjoin Compatible (emptyGraph 3) (fun _ => 1) edge23 and the equality giving edge23's objective as 3/4 with the current universal bound and equality-case uniqueness; these facts are already available from printed_variables_counterexample.", 'second_review': "T23(c) says that the optimum is 3/4 and is attained uniquely by the single edge {2,3}, but P011/P011_Audit.lean:1348 exports only the universal upper bound and the implication from equality to A = edge23. Feasibility and attainment are missing conclusions, so the statement should include them. This is minor: from the explicit edge23 at :1308, its degrees are (0,1,1) and its objective is (1/2)(3/2) = 3/4 by direct substitution; precisely these facts are also stated at :1323. The adjudicator's major severity overstates this elementary finite omission.", 'severity': 'minor'}
Proved AronowAudit.P011.graph_three_cases Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The optimum result already supplies the bound and equality case; the finite graph enumeration is not used separately.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:cc22b4231901c65da0759a562c1a2d431952183668312dba75075d145283cc1a

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Printed statement refuted, p. 2244] "Let $\hat v_{ij}$ be the $ij$th element of the sample covariance matrix with $i = 1, ..., n$ and $j = 1, ..., n$. Since the sample covariance matrix is symmetric, we can focus on its upper triangular part and use the decision variable $a_{ij} = 1$ if $\hat v_{ij} \ne 0$, and 0 otherwise, for each $i < j$." [Counterexample, from the validator] $n = 3$, $\theta_i = 1$, $Y = (-2, \tfrac12, \tfrac32)$, so $\hat\beta = 0$ and the centred weighted outcomes are $(-2, \tfrac12, \tfrac32)$; degree bounds $d = (1, 1, 1)$; no mandatory edges ($\mathcal E_R = \emptyset$). (a) $\hat v_{12} = -1$, $\hat v_{13} = -3$, $\hat v_{23} = \tfrac34$, all nonzero; the printed rule therefore sets $a_{12} = a_{13} = a_{23} = 1$. (b) These values give every vertex degree 2, which violates the constraints $\sum_{j<i}a_{ji} + \sum_{j>i}a_{ij} \le d_i = 1$ of (3.6): the printed "decision variables" are not feasible for the program they are said to define. (c) The optimum of (3.6) on these data is $\tfrac34$, attained only by $a_{23} = 1$, $a_{12} = a_{13} = 0$. (d) The printed rule can also contradict $A \succeq A_R$: an edge $\{i,j\} \in \mathcal E_R$ with $\hat v_{ij} = 0$ gets $a_{ij} = 0$. An instance (constructed here to illustrate the validator's remark): $Y = (0, 1, -1)$, $\theta_i = 1$, $\hat\beta = 0$, $\mathcal E_R = \{\{1,2\}\}$, where $\hat v_{12} = 0$.

: SAP.P011b.T23Statement

Theorem: SAP.P011b.T23

Statement hash: 35f0f8d083887c70d7f2f4ffe1eac5cfff033ed0185161a4988d430a1e18f34a

Statement definition: SAP.P011b.T23Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:54:55Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.graph_three_cases: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T24

Notation corrected in Assumption 2, program (3.5), Assumption 3 and Section 3.2: the observed data $Z$ , the relation $\preceq$ , the index set of Assumption 3, and the edge set $\mathcal E_S$

Assumption 2: p. 2242 (PDF p. 5); program (3.5): p. 2244 (PDF p. 7); Assumption 3 and the last paragraph: p. 2246 (PDF p. 9); Discussion: p. 2248 (PDF p. 11)

Assumption 2, p. 2242; (3.5), p. 2244; Assumption 3, p. 2246; Discussion, p. 2248. Element-wise graph inequalities describe exactly the compatible graphs and admit the true graph under valid constraints.

Historical component credits

Proved AronowAudit.P011.corrected_notation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The Compatible premise is precisely valid true-graph constraints, rather than an extra rank or stochastic restriction. This row covers T24(i), not the separate observed-index correction T24(ii).
Proved AronowAudit.P011.degree_row_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; A row-sum auxiliary already used in the existing feasible-set equivalence proof.
Proved AronowAudit.P011.simple_degree_bound Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A degree auxiliary; it does not supply the omitted statistical and textual clauses.
Proved AronowAudit.P011.compatible_truncated_degree Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A degree auxiliary; the retained claim is the graph feasible-set equivalence.
Proved AronowAudit.P011.program_feasible_iff Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Duplicates the feasible-set equivalence already supplied by corrected_notation.

Formal result: restricted

Scope: Only formal content (i), the graph feasible-set equivalence and true-graph feasibility under valid constraints, is stated. The observed-data tuple correction, the Assumption 3 index-set correction and all of formal content (ii), including its covariance bound and scope of the cited statistical results, and the edge-set typography correction are omitted.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:47825a3fb4c59d96ea867c3d3b809762f9dac34260fd4784bd9a61984b45def3

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected text, from the validator] (a) In Assumption 2 and in the Discussion the observed data are $Z = (Y, d, \theta, \mathcal G_R)$, as defined after Assumption 1 on p. 2240 (the print has $Z = (\mathcal G_R, Y, d)$ on p. 2242 and $Z = (Y, d, \mathcal G_R)$ on p. 2248, omitting $\theta$). (b) After program (3.5): "$\preceq$ denotes the element-wise 'less-than-or-equal-to' relation" (the print has "less-than"). (c) Assumption 3 (Homoskedasticity): $\mathrm{var}(\theta_iY_i) = \mathrm{var}(\theta_jY_j), \forall i, j \in \mathcal V_S$ (the print has $\forall i, j \in \mathcal V$). (d) On p. 2246, "the maximum number of edges that can be in $\mathcal E_S$" (the print has a roman $\mathrm E_S$). [Formal content] (i) With $\preceq$ read as element-wise $\le$ and $\succeq$ as element-wise $\ge$, the feasible set of (3.5) and of (3.8), $\{A \text{ binary symmetric zero-diagonal} : A1 \preceq d,\ A \succeq A_R\}$, is exactly $\mathcal A(Z)$ of Definition 4, and it contains $A^{\mathcal O}$ (valid constraints). (ii) Assumption 3 as corrected is a condition on the $n$ random variables $\theta_1Y_1, \ldots, \theta_nY_n$; the coefficients $\theta_i$ are defined only for $i \in \mathcal V_S = \{1, \ldots, n\}$, while $\mathcal V = \{1, \ldots, N\}$ with $N \ge n$. The bound $\mathrm{cov}(\theta_iY_i, \theta_jY_j) \le \sigma_\theta^2A^{\mathcal O}_{ij}$ for $i \ne j$ in $\mathcal V_S$, the estimators (3.7) and (3.9), Proposition 2, Lemma 1 and Corollaries 2 and 4 involve only $i, j \in \mathcal V_S$.

{n : ℕ} (R G : Graph n) (d : Fin n → ℕ)
    (hv : Compatible R d G) : SAP.P011b.T24Statement R G d hv

Theorem: SAP.P011b.T24

Statement hash: ee600a917def92c8ca4339643cdbf6f99529d81e2167b6a61dbe8ff651d54f0e

Statement definition: SAP.P011b.T24Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:54:55Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.degree_row_sum: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T25

Appendix B, what depvar calculates: the package's variance helpers against formulas (3.1), (3.7) and (3.9)

Appendix B, p. 2250 (PDF p. 13); (3.1): p. 2242 (PDF p. 5); (3.7): p. 2246 (PDF p. 9); (3.9): p. 2247 (PDF p. 10); code: staged/packages/depinf/jrzubizarreta-depinf-f11933a/R/depvar.r, lines 11, 19 and 27

Appendix B, p. 2250; (3.1), p. 2242; (3.7), p. 2246; (3.9), p. 2247; helper code lines 11, 19, 27. Changing the helper divisor from n to n minus one has the stated exact discrepancies, order, maximizing graphs and bound.

Historical component credits

Proved AronowAudit.P011.helper_denominators Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P011.helper_correction Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:3f74b01de9df9b24822ab93d4c343c48c8dd9663d17061eed07ad1c9616f982a

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Corrected claim, from the validator] Appendix B says that the package has "depvar for calculating the variance estimates (3.1) and (3.7)". As staged, the three helpers of depvar use R's var(x), whose divisor is $n - 1$, where the paper defines $\hat\sigma_\theta^2$ with divisor $n$. Fix $n > 1$, $x_i = \theta_iY_i$, $\bar x = \hat\beta$, $z_i = x_i - \bar x$, $S = \sum_iz_i^2$, an adjacency matrix $A$, $C_A = \sum_{i,j}A_{ij}z_iz_j$, $D_A = \sum_{i,j}A_{ij}$ and $D' = \sum_i\min\{d_i, n-1\}$. The printed formulas are $\widehat V_1(A;Z) = \frac{S}{n^2} + \frac{C_A}{n^2}$, $\widehat V_2(A;Z) = \frac{S}{n^2}\left(1 + \frac{D_A}{n}\right)$, $\widehat V_2'(Z) = \frac{S}{n^2}\left(1 + \frac{D'}{n}\right)$; the staged helpers compute $V_1^{R} = \frac{S}{n(n-1)} + \frac{C_A}{n^2}$, $V_2^{R} = \frac{S}{n(n-1)}\left(1 + \frac{D_A}{n}\right)$, $V_2'^{R} = \frac{S}{n(n-1)}\left(1 + \frac{D'}{n}\right)$. Then: (i) $V_1^{R} - \widehat V_1(A;Z) = \frac{S}{n^2(n-1)} = \frac{\hat\sigma_\theta^2}{n(n-1)}$ (only the first term of (3.1) is rescaled); (ii) $V_2^{R} = \frac{n}{n-1}\widehat V_2(A;Z)$ and $V_2'^{R} = \frac{n}{n-1}\widehat V_2'(Z)$; (iii) each code value is at least the printed one, strictly when $S > 0$; and for fixed data a matrix maximizes $V_1^{R}$ (respectively $V_2^{R}$) over $\mathcal A(Z)$ if and only if it maximizes $\widehat V_1$ (respectively $\widehat V_2$); (iv) if $|x_i - \beta| \le c_1$ for all $i$, then $0 \le n\left(V_1^{R} - \widehat V_1(A;Z)\right) \le \frac{4c_1^2}{n-1}$ for every $A$; (v) with var(x) replaced by mean((x - mean(x))^2), i.e. by $S/n$, the three helpers equal (3.1), (3.7) and (3.9).

{n : ℕ} (hn : 1 < n) (R : SAP.P011b.Graph n) (d : Fin n → ℕ)
    (x : Fin n → ℝ) : SAP.P011b.T25Statement hn R d x

Theorem: SAP.P011b.T25

Statement hash: e13b71ef8f4809390b148c0c06d64501dbf067dc0b57aca5772ceb3b40603022

Statement definition: SAP.P011b.T25Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:52:54Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T26

Appendix B, "depvar for calculating the variance estimates (3.1) and (3.7)", against the staged helpers

Appendix B, p. 2250 (PDF p. 13); code: staged/packages/depinf/jrzubizarreta-depinf-f11933a/R/depvar.r, lines 11, 19 and 27

The printed claim is false: Three exact data vectors separate the helper formulas from the paper formulas, and one occurs with positive probability in an admissible family. Appendix B, p. 2250; helper code lines 11, 19, 27.

Historical component credits

Counterexample proved AronowAudit.P011.helper_counterexamples Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is an exact witness certificate for the two sets of formulas supplied in the target, not an execution theorem about R; R076-04 and R076-05 supply the statistical admissibility of witness (a).
Proved AronowAudit.P011.helper_witness_law Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P011.helper_witness_scaling Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:2fcc7f22e533e91aace29ba680d3f45ef69d8bb6f980acd254d7452fe7e0a73a

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [Statement refuted, p. 2250] "depinf includes two basic functions: depgraph, for finding an adjacency matrix that maximizes the estimated variance of $\hat\beta$ given general constraints on the degree of dependence of the observations (these are problems (3.5) and (3.8) above), and depvar for calculating the variance estimates (3.1) and (3.7)." [Exact witnesses, from the validator; notation of P011-T25] (a) $n = 3$, $x = (0, 1, 2)$, $\theta = (1, 1, 1)$, $A = 0$, $d = (0, 0, 0)$: $\bar x = 1$, $S = 2$, $C_A = D_A = D' = 0$. The printed formulas give $\widehat V_1 = \widehat V_2 = \widehat V_2' = 2/9$; the staged helpers give $1/3$ for each. (b) $n = 3$, $x = (0, 0, 3)$, $\theta = (1, 1, 1)$, $A = \begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}$, $d = (1, 1, 0)$: $\bar x = 1$, $z = (-1, -1, 2)$, $S = 6$, $C_A = 2$, $D_A = D' = 2$. Printed: $\widehat V_1 = 8/9$, $\widehat V_2 = \widehat V_2' = 10/9$. Staged helpers: $11/9$, $5/3$, $5/3$. The ratio is $11/8$ for $V_1$ and $3/2$ for $V_2$ and $V_2'$, so $V_1$ is not rescaled as a whole. (c) $n = 3$, $x = (0, 1, 2)$, $A$ the complete graph on three vertices: $C_A = -S = -2$, so the printed $\widehat V_1 = 0$ while the staged helper gives $1/9$; the relative difference of $V_1$ is not bounded. (d) The data of (a) are possible under a model satisfying Assumptions 2 and 3: independent outcomes uniform on $\{0, 1, 2\}$, $\theta_i = 1$, empty dependency graph, $\beta = 1$, $n\,\mathrm{var}(\hat\beta) = 2/3$; the observed triple has probability $1/27$.

: SAP.P011b.T26Statement

Theorem: SAP.P011b.T26

Statement hash: 2d61edced2c20e2fd36ad6dae4917db9f37070817a0d5aa01743a50a60c5d180

Statement definition: SAP.P011b.T26Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:52:54Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T27

Definitions 4–6 with the sharp upper bound for the oracle estimator, and the integer programs (3.5), (3.6) and (3.8)

Definitions 4–6 and the sharp-bound sentence: p. 2241 (PDF p. 4); program (3.5): p. 2244 (PDF p. 7); program (3.6): p. 2245 (PDF p. 8); program (3.8): p. 2246 (PDF p. 9)

Definitions 4–6, p. 2241; (3.5), p. 2244; (3.6), p. 2245; (3.8), p. 2246. Compatible variance maximizers exist, bound the oracle sharply, and are characterized by the stated finite integer programs.

Historical component credits

Proved AronowAudit.P011.oracle_sharp_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Compatibility is the definition of valid observed constraints in S10(h), not an extra statistical assumption; existence of a compatible graph is thus explicitly nonvacuous. No claim of domination of the true variance is made.
Proved AronowAudit.P011.integer_program_estimators Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The opening row label says (c)–(e), but its next sentence explicitly assigns the upper-triangular part (d) elsewhere; I therefore assess this row as (c),(e), leaving (d) uncovered in this chunk.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:1ad05e4dc1eeaa3b804f0d1061592fd3998d8d747b74a5999eb547d4ee4fa9d4

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2241] "it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "The maximal compatible estimator provides a sharp upper bound for the oracle estimator because $\widehat V(A^{\mathcal O};Z) \le \widehat V(A^m;Z)$, with equality when $A^{\mathcal O} \in \mathcal A^m$." [As printed, p. 2244] "We can find an element $A^m$ of $\mathcal A_1^m$ by solving the 0-1 integer linear program maximize over $A$: $(\Theta Y - \hat\beta)'A(\Theta Y - \hat\beta)$, subject to $A1 \preceq d$, $A \succeq A_R$, (3.5) where $A_R$ is the adjacency matrix of $\mathcal G_R$ and $\preceq$ denotes the element-wise "less-than" relation. Since $A$ is an adjacency matrix, we can reduce the program and maximize over the decision variables that correspond to the upper or lower triangular elements of $A$ only." [As printed, p. 2245] "the integer linear program (3.5) can be written as maximize over $\boldsymbol a$: $\sum_{i=1}^{n}\sum_{j=i+1}^{n}\hat v_{ij}a_{ij}$, subject to $\sum_{j=1}^{i-1}a_{ji} + \sum_{j=i+1}^{n}a_{ij} \le d_i,\ i = 1, ..., n$, and $a_{ij} \in \{0, 1\},\ i = 1, ..., n,\ j = 1, ..., n,\ i < j$, (3.6) where $d_i$ is the degree, and further simplified with the constraints $A \succeq A_R$ that make some of the decision variables $a_{ij}$ automatically equal to one. The resulting program has at most $n(n-1)/2$ decision variables". [As printed, p. 2246] "To find an element of $\mathcal A_2^m$, we solve the 0-1 integer linear program maximize over $A$: $1'A1$, subject to $A1 \preceq d$, $A \succeq A_R$, (3.8) where again $A$ is an arbitrary 0-1 adjacency matrix and $A_R$ is the adjacency matrix of $\mathcal G_R$. In order to solve the program (3.8), let $\hat v_{ij} = 1$ for every $i = 1, ..., n$ and $j = 1, ..., n$ in (3.6)." [The claims, for every $n$ and every realization of the data, with the constraints valid in the sense of (S10)(h)] (a) $A^{\mathcal O} \in \mathcal A(Z)$. (b) $\mathcal A(Z)$ is finite and nonempty, so for any real-valued $\widehat V(\cdot;Z)$ on $\mathcal A(Z)$ a maximizer $A^m$ exists, and $\widehat V(A^{\mathcal O};Z) \le \widehat V(A^m;Z)$, with equality when $A^{\mathcal O} \in \mathcal A^m$. (c) $A$ is feasible for (3.5) if and only if $A \in \mathcal A(Z)$, and $A$ solves (3.5) if and only if $A \in \mathcal A_1^m$, because $n^2\widehat V_1(A;Z) = n\hat\sigma_\theta^2 + (\Theta Y - \hat\beta 1)^{\mathsf T}A(\Theta Y - \hat\beta 1)$. (d) (3.6), with $\hat v_{ij} = (\theta_iY_i - \hat\beta)(\theta_jY_j - \hat\beta)$, $a_{ij}$ the indicator that the edge $\{i,j\}$ is selected, and $a_{ij} = 1$ for $\{i,j\} \in \mathcal E_R$, has the same solutions as (3.5) under $A_{ij} = A_{ji} = a_{ij}$, and has at most $n(n-1)/2$ decision variables. (e) Every solution of (3.8) is in $\mathcal A_2^m$, and conversely when $\hat\sigma_\theta^2 > 0$, because $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac1n 1^{\mathsf T}A1\right]$; (3.8) is (3.6) with every $\hat v_{ij} = 1$.

{n : ℕ} (hn : 0 < n) (R G : SAP.P011b.Graph n) (d : Fin n → ℕ)
    (hv : SAP.P011b.Compatible R d G) (x : Fin n → ℝ) :
    SAP.P011b.T27Statement hn R G d hv x

Theorem: SAP.P011b.T27

Statement hash: f24f587e47a814cb9d51d273238ce31adc1daf15a6e627523346f36b811b6f31

Statement definition: SAP.P011b.T27Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:52:54Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T28

Lemma 1

p. 2247 (PDF p. 10); proof in Appendix A, pp. 2249–2250 (PDF pp. 12–13)

Lemma 1, p. 2247; Appendix A, pp. 2249–2250. The maximal homoskedastic estimator lies between the oracle and the truncated-degree bound, with equality when every cap is attained.

Historical component credits

Proved AronowAudit.P011.lemma_one Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:b586f64bd3342026c33462b5557924da282b9a16bea7f3c16c25e04aaa9aaf83

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2247] "Lemma 1. Let $A^m \in \mathcal A^m$. Then $\widehat V_2(A^{\mathcal O}, Z) \le \widehat V_2(A^m;Z) \le \widehat V_2'(Z)$, with $\widehat V_2(A^m;Z) = \widehat V_2'(Z)$ when $d_i' = \sum_{j \in \mathcal V_S}A^m_{ij}$ for each $i \in \mathcal V_S$." [Hypotheses attached] $A^m$ is a maximizer of $\widehat V_2(A;Z)$ over $\mathcal A(Z)$ (the set $\mathcal A_2^m$ of p. 2246); $d_i' = \min\{d_i, n-1\}$; the constraints are valid, so that $A^{\mathcal O} \in \mathcal A(Z)$. The statement holds in every realization and for every $n$; it does not use Assumption 2 or Assumption 3.

{n : ℕ} (hn : 0 < n) (R G A : Graph n) (d : Fin n → ℕ)
    (hv : Compatible R d G) (x : Fin n → ℝ)
    (hm : Maximizes R d (fun B => V2 B x) A) : SAP.P011b.T28Statement hn R G A d hv x hm

Theorem: SAP.P011b.T28

Statement hash: 5e93252beb8043276b8e66c295dd9b8f1d918b662b9984c0d2b7eee75e29ffd6

Statement definition: SAP.P011b.T28Statement

Certificate. 20261104T191443Z

Declaration review: faithful; 2026-11-04T18:52:54Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T29

Section 2: ordinary least squares coefficients as linear unbiased estimators, and the variance of a linear estimator

Definition 3 and the regression example: pp. 2240–2241 (PDF pp. 3–4); the variance display: p. 2241 (PDF p. 4)

Definition 3 and Section 2, pp. 2240–2241. OLS coefficients have the stated linear representation and are unbiased under an integrable linear mean model with invertible Gram matrix, and any linear estimator has the stated covariance-sum variance under finite second moments and positive sample size.

Historical component credits

Proved AronowAudit.P011.linear_variance_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The finite second moments are in target T29(c) itself, so MemLp is not an added restriction; arbitrary underlying probability spaces can be replaced by the Borel joint outcome law without changing these quantities.
Proved AronowAudit.P011.ols_linear_unbiased Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Integrability expresses the meaningful finite-coordinate expectation assumption of E[Y]=Db; nonzero Gram determinant is the target’s explicit invertibility premise. Allowing p=0 does not exclude any positive-dimensional case.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:e00786e2a60d901577a24cb733c24690ae4438f2c1af830526b8c6d3bee49d59

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2240] "Many familiar linear estimators are unbiased in settings with outcomes that are neither independent nor identically distributed. For example, consider the linear regression model $\mathrm{E}[Y_i] = \sum_{j=1}^{p}x_{ij}\beta_j$ for $i = 1, \ldots, n$ [e.g., 10, Ch. 1], where $x_i$ and $\beta$ have dimension $p \times 1$, with the coefficients $\beta$ estimated by ordinary least squares. Let $X$ be the $n \times p$ matrix of covariates, and let $Y$ be the $n \times 1$ vector of outcomes. The estimated coefficients are $\hat\beta = (X'X)^{-1}X'Y$." [p. 2241] "Define the $p \times n$ matrix $\Theta = n(X'X)^{-1}X'$. Then for $j = 1, \ldots, p$ we can express the vector coefficient estimates as $\hat\beta = n^{-1}\Theta Y$, or individually as $\hat\beta_j = \frac1n\sum_{i=1}^{n}\Theta_{ji}y_i$, as in Definition 3." "$\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n}\theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." [The claims] For a fixed $n \times p$ matrix $X$ with $X'X$ invertible: (a) $\hat\beta_j = \frac1n\sum_{i=1}^{n}\Theta_{ji}Y_i$ for $j = 1, \ldots, p$; (b) if $\mathrm{E}[Y] = X\beta$ then $\mathrm{E}[\hat\beta_j] = \beta_j$, so $\hat\beta_j$ is a linear unbiased estimator of $\beta_j$ in the sense of Definition 3 with $\theta_i = \Theta_{ji}$, whatever the dependence among the $Y_i$ and whether or not they are identically distributed; (c) for fixed coefficients $\theta_1, \ldots, \theta_n$ and outcomes with finite second moments, $\mathrm{var}\left(\frac1n\sum_i\theta_iY_i\right) = \frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$.

: SAP.P011c.T29Statement

Theorem: SAP.P011c.T29

Statement hash: a6df5509bbb6786c5e879cb55d982b626794f6ed6995e7f65415b9e4876e178f

Statement definition: SAP.P011c.T29Statement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:37:17Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T30

The computational classification of program (3.5)/(3.6), as printed

p. 2245 (PDF p. 8), the two paragraphs after program (3.6)

program (3.6) and its computational classification, p. 2245. For positive sample size, the degree-constrained unordered-edge program has exactly the same maximizers as the two-endpoint incidence 0/1 knapsack with the same mandatory edges and empirical edge profits; the variance objective is their positive affine normalization. Restricted to C1's optimization encoding: omit C2's NP-hardness assertion or its correction, C3's PTAS assertion, and running-time and decision-variable-count assertions. No algorithm or hardness transfer from general knapsack is asserted.

Historical component credits

Proved AronowAudit.P011.knapsack_encoding Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Only the auxiliary feasibility encoding is claimed here; the rational weight field is unused, so it restricts no case of this incidence-count identity. No optimization-level objective correspondence, NP-hardness or PTAS is exported.

Formal result: restricted

Scope: Retain only C1's optimization encoding in program (3.6)'s unordered-edge formulation: exactly the same maximizers for the degree-constrained normalized variance objective and the two-endpoint incidence 0/1 knapsack with the same mandatory edges and empirical real edge profits. Require positive sample size for the normalized objective. Omit C2 (NP-hardness or its correction), C3 (PTAS), running-time and decision-variable-count assertions. No algorithm or transfer of hardness from general knapsack is asserted.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:86658669c476bd8d3d942084a2ab86844b53380d968b651f5f825b32d2d847e8

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2245] "The resulting program has at most $n(n-1)/2$ decision variables and in general it is a multidimensional knapsack problem [12]. See chapter 9 of Kellerer et al 13 for an overview of the multidimensional knapsack problem. This program (3.5) is NP-hard, but it admits a polynomial time approximation scheme (PTAS). Typical PTAS depend heavily on the size of the problem and their running time can be very high (see, e.g., section 9.4.2 of Kellerer et al 12)." [The three claims] (C1) program (3.6) is, in general, a multidimensional knapsack problem; (C2) program (3.5) is NP-hard; (C3) it admits a polynomial-time approximation scheme. [Hypotheses attached] The program is (3.5), equivalently (3.6): maximization over the compatible adjacency matrices of Definition 4, with vertex degree upper bounds $d_i$ and the mandatory edges of $\mathcal G_R$; no other constraint on the graph is part of it.

: SAP.P011c.T30Statement

Theorem: SAP.P011c.T30

Statement hash: 612a327771962cf721d6ca52df8e45b4f4c01a3e546dab3921ea69c3162bb857

Statement definition: SAP.P011c.T30Statement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:37:17Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T31

Proposition 1

p. 2242 (PDF p. 5); proof pp. 2242–2244 (PDF pp. 5–7)

The printed claim is false: P011-T31; Proposition 1, p. 2242. The universal n-scaled consistency guarantee for the general oracle estimator fails under the printed assumptions.

Historical component credits

Counterexample proved AronowAudit.P011.Wave1.printed_proposition_one_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Positive growing-size reindexing is a lossless way to test the asserted universal n-limit: failure along nested independent path blocks contradicts it. Joint outcome laws represent arbitrary underlying spaces; the statement does not assume the convergence it negates. The record did not change my claim-stage verdict.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:d5a5515a48fb7e5017143a25a9f17caf695e8f71983aa2bf460d21688475167b

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2242] "The corresponding oracle estimator $\widehat V_1(A^{\mathcal O};Z)$ is consistent. Proposition 1. Under Assumption 2, for any $\epsilon > 0$, $\lim_{n\to\infty}\Pr(|n\widehat V_1(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) = 0$." [Hypotheses attached] The setting of Section 2 (Definitions 1–3, Assumption 1): $\mathcal G$ a dependency graph, $\hat\beta = \frac1n\sum_{i \in \mathcal V_S}\theta_iY_i$ with $\mathrm{E}[\hat\beta] = \beta$; $A^{\mathcal O}$ the adjacency matrix of the induced subgraph $\mathcal G_S$; $\widehat V_1$ as in (3.1); Assumption 2. No condition on the individual means $\mathrm{E}[\theta_iY_i]$ is attached.

: SAP.P011c.T31Statement

Theorem: SAP.P011c.T31

Statement hash: 0303e1efdbac14e63ff8f378d48522677634fe08474f5536c168fcac02565690

Statement definition: SAP.P011c.T31Statement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:37:17Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T32

Corollary 1, with its proof (A.1) and the sentence after Assumption 3

Corollary 1: p. 2245 (PDF p. 8); proof: Appendix A, p. 2249 (PDF p. 12); sentence after Assumption 3: p. 2246 (PDF p. 9)

The printed claim is false: P011-T32; Corollary 1, p. 2245, and its homoskedastic specialization on p. 2246; proof (A.1), p. 2249. The combined universal conservativeness contract fails; this negation does not separately assert that its homoskedastic specialization fails.

Historical component credits

Counterexample proved AronowAudit.P011.Wave1.printed_corollary_one_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': 'P011/Wave1/CorollaryOne.lean:27 negates the universal contract whose conclusion is C AND (H implies C), with C exactly the printed n-scaled conservativeness assertion. That conjunction is logically equivalent to C, so its negation genuinely refutes the general Corollary 1 guarantee under the stated scaling, valid-constraint and maximization hypotheses. It does not separately certify a homoskedastic counterexample, but the current row description accurately identifies the redundant specialization and claims the combined negation.'}
Proved AronowAudit.P011.underestimation_event_inclusion Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A valid finite proof component, not needed to transport the negation of the universal printed guarantee.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:3d0c3c2f4097eb999ba37689b295c10f0730bd64b2dbdcc5bdea5423fe1ae405

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2245] "While the true adjacency matrix $A^{\mathcal O}$ is not known, an element $A^m \in \mathcal A_1^m$ produces a variance estimate $\widehat V_1(A^m, Z)$ that is at least as large as the oracle estimator $\widehat V_1(A^{\mathcal O};Z)$. As $n$ grows large, the variance estimate $\widehat V_1(A^m, Z)$ is conservative: the probability that $n\widehat V_1(A^m)$ underestimates $n\,\mathrm{var}(\hat\beta)$ by more than $\epsilon > 0$ tends to zero. Corollary 1. Given Assumption 2, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_1(A^m;Z) > \epsilon) = 0$ for any $\epsilon > 0$." [As printed, p. 2246] "Under homoskedasticity, the general estimator $\widehat V_1(A^m, Z)$ developed in Section 3.1 provides a conservative variance estimate." [Hypotheses attached] The setting of Section 2; $A^m \in \mathcal A_1^m$, a maximizer of $\widehat V_1(A;Z)$ over the compatible set $\mathcal A(Z)$ defined by the known edges $\mathcal E_R$ and the degrees $d_i$; Assumption 2 (and Assumption 3 for the sentence on p. 2246). No condition on the individual means $\mathrm{E}[\theta_iY_i]$ and no restriction on $\mathcal E_R$ is attached.

: SAP.P011c.T32Statement

Theorem: SAP.P011c.T32

Statement hash: 95fd14bfadf051b609c6541251d97aeab4a641c83f495b8191d50b676a547ef3

Statement definition: SAP.P011c.T32Statement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:37:17Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T33

Corollary 3

p. 2248 (PDF p. 11)

Corollary 3, p. 2248. The universal nominal asymptotic coverage guarantee fails under the explicit convention that negative-estimate realizations, whose printed intervals are undefined, count as misses. Restricted to this extension of the partial printed interval: the proof does not refute the guarantee on its real-square-root domain or establish undercoverage for estimates that are nonnegative almost surely. No assertion about such defined intervals is added.

Historical component credits

Counterexample proved AronowAudit.P011.Wave1.printed_corollary_three_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P011/Wave1/CorollaryThree.lean:26 negates the target's universal coverage guarantee under scaling, valid constraints and V1 maximization. PrintedConservativeCoverage uses the explicit nonnegative-estimate event in P011/P011_Audit.lean:235, counting realizations on which the printed real square root is undefined as uncovered. This gives the stated printed interval a coverage event consistent with S10(d); it neither assumes positivity nor substitutes the corrected clipped interval."}
Proved AronowAudit.P011.interval_containment Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A finite interval comparison, not needed separately for the retained target conclusion.
Proved AronowAudit.P011.printed_coverage_eq_of_nonnegative Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A finite nonnegative-estimate comparison, not needed separately for this target conclusion.

Formal result: restricted

Scope: Retain the negation of the universal coverage guarantee only under the explicit extension that negative-estimate realizations, whose printed real intervals are undefined, count as uncovered. The existing path witness has n*V1 tending to -11/15 and uses this convention. It does not refute the guarantee on its real-square-root domain or establish undercoverage when V1 is nonnegative almost surely; those conclusions are not claimed. The Lean statement and existing proof are unchanged.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:e1b3ff595434ab79ccffe7b2e94e359f44931f9761539443e4922a06fbb08860

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2248] "It follows that Wald-type confidence intervals constructed using the conservative variance estimators derived in Section 3 yield conservative asymptotic coverage. Corollary 3. Given Assumption 2, confidence intervals formed as $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V_1(A^m)}$ have at least $100(1-\alpha)\%$ coverage for $\beta$ in large $n$." "Proofs for Corollaries 3 and 4 follow directly from Corollaries 1 and 2 and Proposition 3." [Hypotheses attached] The setting of Section 2; $A^m \in \mathcal A_1^m$; Assumption 2. Read as: for every $0 < \alpha < 1$, $\liminf_{n\to\infty}\Pr\left(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V_1(A^m;Z)}\right) \ge 1 - \alpha$.

: SAP.P011c.T33Statement

Theorem: SAP.P011c.T33

Statement hash: 1b13fe0d1be7b7eb6c0572592b8b43e018069162798a9e68e2bb7906e5a0fa82

Statement definition: SAP.P011c.T33Statement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:37:17Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T34

Proof of Proposition 2, as printed

p. 2246 (PDF p. 9)

The printed claim is false: P011-T34; proof of Proposition 2, p. 2246. Independent fair signs with unit weights and empty dependency graphs satisfy Assumptions 2 and 3, valid constraints, and even the stronger observed-graph nesting. Their sample sizes are k+1, their target means are zero, and their scaled estimator variances are one. For every choice of the undefined one-index coefficients and every positive epsilon, the printed auxiliary estimator's less-than-epsilon probability tends to one and does not tend to zero. This refutes the printed argument through its P4 assertion without assigning a meaning to the undefined correlations or repairing P2.

Historical component credits

Proved AronowAudit.P011.printed_auxiliary_degree_bound Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; This row covers P3 alone, exactly as described; the other printed proof steps remain outside its conclusion. The generalization dropping homoskedasticity is valid and harmless.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:095b11f70ab2eefca18d99d3435fc0a16597c93de18ce6f052caaeee3fdc5baf

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2246] "Proof. We first define an alternative oracle estimator which presumes knowledge of the correlations $\rho_i$, $\widehat V_2^*(A^{\mathcal O};Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij}\rho_i\right]$. Multiplying by $n$, $n\widehat V_2^*(A^{\mathcal O};Z) = \hat\sigma_\theta^2\left[1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij}\rho_i\right]$. As in the proof of Proposition 1, $\hat\sigma_\theta^2$ converges in mean square. By Assumption 2, $1 \le 1 + \frac1n\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S}A^{\mathcal O}_{ij} \le 1 + c_2$, allowing us to invoke Slutsky's Theorem and Chebyshev's Inequality to show $\lim_{n\to\infty}\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| < \epsilon) = 0$. The Cauchy-Schwarz Inequality (i.e., all $\rho_i \le 1$) implies $\widehat V_2^*(A^{\mathcal O};Z) \le \widehat V_2(A^{\mathcal O};Z)$ across all sample realizations. The result follows directly." [The steps asserted] (P1) the definition of $\widehat V_2^*$ with "correlations $\rho_i$"; (P2) $\hat\sigma_\theta^2$ converges in mean square; (P3) $1 \le 1 + \frac1n\sum_{i}\sum_{j}A^{\mathcal O}_{ij} \le 1 + c_2$; (P4) $\lim_{n\to\infty}\Pr(|n\widehat V_2^*(A^{\mathcal O};Z) - n\,\mathrm{var}(\hat\beta)| < \epsilon) = 0$; (P5) all $\rho_i \le 1$, hence $\widehat V_2^*(A^{\mathcal O};Z) \le \widehat V_2(A^{\mathcal O};Z)$ in every realization. [Hypotheses attached] Assumptions 2 and 3; the setting of Section 2.

: SAP.P011c.T34WholeStatement

Theorem: SAP.P011c.T34

Statement hash: 819e3c08902633841630affd85d58dedda81c5704883d37cdfc9190487c7ffd3

Statement definition: SAP.P011c.T34WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:39:49Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.printed_auxiliary_degree_bound: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T35

The package's claim of conservative variance estimates for generic linear unbiased estimators, as applied in the vignette to OLS coefficients with known recruitment edges

staged/packages/depinf/jrzubizarreta-depinf-f11933a/README.md, line 3; vignettes/replication.Rmd, lines 15 and 100–105; the article's Appendix B, p. 2250 (PDF p. 13), describes the package

The printed claim is false: P011-T35; package README line 3 and vignette lines 15, 100–105; Appendix B, p. 2250. There is a correctly specified two-column OLS coefficient problem with coefficient vector (0,1), invertible designs, valid known-edge and degree constraints, and Assumption 2 constants (6,2,29/15,2), whose maximizing divisor-n general estimator fails n-scaled variance conservativeness. The witness even has nested observed graphs, which can serve as the full graphs. Thus the combined formal variance and printed Wald-coverage guarantee is false. These are mathematical counterexample facts; they make no assertion about the vignette's realized standard errors.

Historical component credits

Counterexample proved AronowAudit.P011.Wave1.printed_package_ols_guarantee_refuted Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P011/Wave1/PackageOLS.lean:29 negates precisely the interpreted T35 guarantee of both V1 conservativeness and printed-interval coverage for correctly specified, full-rank OLS designs. Refuting their conjunction requires failure of at least one guarantee, which is sufficient to refute this combined claim and does not assert two separate counterexamples. The fixed-dimensional design, actual mean-model condition and n-scaled OLS weights match the target, and the row claims nothing about the vignette's realized standard errors."}
Proved AronowAudit.P011.printed_coverage_eq_of_nonnegative Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A finite nonnegative-estimate comparison, not needed separately for this target conclusion.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f8e8f7ed26bb3586794aa2e5c0262cf8556b7ab1abbdeecd218c633889929e20

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As written in the package README, line 3] "This package implements the methods for constructing confidence intervals for linear unbiased estimators given constraints on units' dependency graph described in Aronow, Crawford and Zubizarreta (n.d.). The primary function, depvar, produces conservative variance estimates for generic linear unbiased estimators" [As written in the vignette] Line 15: "In this vignette, we replicate the main analyses in Section 5 of the paper." Lines 100–101: "# V1, some known dependencies" and "b1 = sqrt(depvar(Y, Theta[1, ]*n, d, GR, case = "heteroskedastic", solver = "gurobi", approximate = 0)\$V_hat)", with the analogous calls for coefficients 2 to 4. [Formal reading] For an OLS coefficient $\hat\beta_j = \frac1n\sum_i\Theta_{ji}Y_i$ under the linear model $\mathrm{E}[Y_i] = x_i'\beta$ of p. 2240, with $\theta_i = \Theta_{ji}$, known recruitment edges $\mathcal E_R$ and degrees $d_i$, and $A^m \in \mathcal A_1^m$: given Assumption 2, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta_j) - n\widehat V_1(A^m;Z) > \epsilon) = 0$ for any $\epsilon > 0$ (Corollary 1), and the Wald interval of Corollary 3 has at least nominal coverage in large $n$.

: SAP.P011c.T35WholeStatement

Theorem: SAP.P011c.T35

Statement hash: 78c47cd59e73c7762b3c9a355e642e88140a06af09c1e8bed61241b9641bbbde

Statement definition: SAP.P011c.T35WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:39:49Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.Wave1.printed_package_ols_guarantee_refuted: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T36

Proposition 2

p. 2246 (PDF p. 9)

Proposition 2 and its introductory sentence, p. 2246. Under Assumptions 2 and 3 the homoskedastic oracle estimator is asymptotically conservative, with arbitrary individual weighted means. No relation between the observed graphs is required. It need not be consistent in the surrounding n-scaled sense: there is a family with constants (2,1,1,2), even with nested observed graphs, for which n times the oracle estimate does not converge in probability to n times the true variance. This states the relative/scaled meaning of consistency used for variance inference here.

Historical component credits

Proved AronowAudit.P011.proposition_two Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The row proves the displayed Proposition 2. The target also quotes an introductory nonconsistency assertion, which is not established by this conservativeness conclusion alone; see targets.json.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:5a2e9589583492f22afef00c796fd3f949ed287d9bdb1360b4b7ceb629c4f4bf

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2246] "The oracle estimator $\widehat V_2(A^{\mathcal O}, Z)$ need not be consistent, though it is asymptotically conservative. Proposition 2. Given Assumptions 2 and 3, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^{\mathcal O};Z) > \epsilon) = 0$ for any $\epsilon > 0$." [Hypotheses attached] The setting of Section 2; $\widehat V_2$ as in (3.7); Assumption 2; Assumption 3 (equal variances of the weighted outcomes $\theta_iY_i$, for the observed units). No condition on the individual means $\mathrm{E}[\theta_iY_i]$.

: SAP.P011c.T36WholeStatement

Theorem: SAP.P011c.T36

Statement hash: 22572f86adfd99a65170f4644e8247a72c56602fbc0439f960a7888c0cae4326

Statement definition: SAP.P011c.T36WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:39:49Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.proposition_two: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T37

Corollary 2

p. 2247 (PDF p. 10)

Corollary 2, p. 2247. Under Assumptions 2 and 3 and valid graph constraints, both the maximizing homoskedastic estimator and the degree-only estimator are asymptotically conservative. The supplied degree caps are unrestricted, individual weighted means can differ, and no relation between observed graphs or measurable choice of maximizing graphs is required.

Historical component credits

Proved AronowAudit.P011.corollary_two Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The law representation, full disconnected-set independence, valid constraints and maximization preserve the claim’s cases; supplied caps are not required to be uniformly bounded. No measurable selector premise is imposed.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:b09eb86ddc96aebf72c71337b549f102739b43d74d3bc9b22b41f4d6b1313eab

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2247] "The upper bound estimators under homoskedasticity are asymptotically conservative. Corollary 2. Given Assumptions 2 and 3, then for any $\epsilon > 0$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2(A^m;Z) > \epsilon) = 0$, $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V_2'(Z) > \epsilon) = 0$." "The proof follows from Lemma 1 and the same reasoning employed in the proof of Corollary 1." [Hypotheses attached] The setting of Section 2; $A^m \in \mathcal A_2^m$; $\widehat V_2$ as in (3.7), $\widehat V_2'$ as in (3.9); valid constraints; Assumptions 2 and 3.

: SAP.P011c.T37WholeStatement

Theorem: SAP.P011c.T37

Statement hash: f169531db795c03a3bc7370c6bed4a6f0b3364da8f3c23da6263b111f6d2a000

Statement definition: SAP.P011c.T37WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:39:49Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.corollary_two: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T38

Lemma 2

p. 2247 (PDF p. 10)

Lemma 2, p. 2247. Under Assumption 2 the estimator has eventually positive variance and its standardized cdf converges at every real threshold to the standard normal cdf. The conclusion holds for arbitrary positive sample sizes tending to infinity and rowwise dependency graphs, and hence in the printed setting of nested full graphs and arbitrary observed subsets. No equality of the individual means is assumed.

Historical component credits

Proved AronowAudit.P011.dependency_graph_clt Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; All real cdf arguments are quantified, giving the usual convergence in distribution because Phi is continuous. Zero early denominators are harmless, since the scaling limit is strictly positive. There is no published-theorem hypothesis to check.
Proved AronowAudit.P011.clt_finite_normalization Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Auxiliary finite normalization; the primary declaration already establishes the complete CLT.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:e443c05fb991b58b140a6c47e63e4e737b18aa7696525557d6b068376aa63245

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2247] "Lemma 2. Given Assumption 2, $\left(\hat\beta - \beta\right)\Big/\sqrt{\mathrm{var}(\hat\beta)} \to_d N(0, 1)$." "Lemma 2, a standard result in applying Stein's method to the setting of local dependence, has been proven by, e.g., Theorem 2.7 of Chen et al [6]." [Hypotheses attached] The setting of Section 2 ($\mathcal G$ a dependency graph, $\hat\beta = \frac1n\sum_{i \in \mathcal V_S}\theta_iY_i$, $\beta = \mathrm{E}[\hat\beta]$) and Assumption 2: $|\theta_iY_i - \beta| \le c_1$ almost surely, degrees in $\mathcal G_S$ at most $c_2$, $n\,\mathrm{var}(\hat\beta) \to c_3 > 0$, $|\theta_i| < c_4$. $\to_d$ is convergence in distribution as $n \to \infty$.

: SAP.P011c.T38WholeStatement

Theorem: SAP.P011c.T38

Statement hash: 4e5729071322af9d6667269fcdbf5c71d988ffa0de4d5480ee807827dd7035ab

Statement definition: SAP.P011c.T38WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:39:49Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.dependency_graph_clt: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T39

Lemma 3 and Proposition 3

Lemma 3: pp. 2247–2248 (PDF pp. 10–11); Proposition 3: pp. 2247–2248 (PDF pp. 10–11), proof p. 2248 (PDF p. 11)

Lemma 3 and Proposition 3, pp. 2247–2248. Under Assumption 2, scaled consistency gives exact limiting coverage and scaled one-sided conservativeness gives at least nominal limiting inferior coverage, for every level strictly between zero and one. Intervals use nonnegative clipping. Statistics may be measurable for the completed joint law, represented by a.e. measurable real functions on the raw outcome space. This includes all a.e. versions. No relation between successive observed graphs or monotonicity of sample sizes is required.

Historical component credits

Proved AronowAudit.P011.conditional_wald_coverage Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P011_CoverageCompletion.lean:349 requires Borel-measurable V on the canonical raw-outcome space, but this does not materially exclude the completed-law-measurable statistics in the reviewer's example. Every such real statistic has a Borel representative equal almost surely: see Mathlib/MeasureTheory/Measure/AEMeasurable.lean:377 and MeasureSpaceDef.lean:435. Replacing each V_k by that representative preserves every consistency, conservativeness and coverage-event probability at each k, so all the T39 conclusions apply without losing any probabilistic case. The measurability premise is therefore a representation convention, and the clipping, quantification over every positive epsilon and two coverage conclusions match T39."}
Proved AronowAudit.P011.truncation_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The scale n is any nonnegative real, so it includes every sample-size specialization needed by the target. Negative V is permitted; nonnegative true variance is not needed for this algebraic step.
Proved AronowAudit.P011.clipped_truncation Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Auxiliary clipping comparison already incorporated in the primary conditional coverage theorem.
Proved AronowAudit.P011.interval_containment Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary event containment for the corrected clipped intervals. For 0<alpha<1 the target’s normal critical value meets hz. Allowing negative U,V with clipping does not remove any intended case.
Proved AronowAudit.P011.printed_coverage_eq_of_nonnegative Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; A finite nonnegative-estimate comparison, not needed separately for this target conclusion.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:986833f19e5e346013b46eb339c6d342eaf010e45c1c12a6b04a1bc58f748ae0

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2247] "Lemma 3. Given Assumption 2, if a variance estimator $\widehat V(A;Z)$ satisfies $\lim_{n\to\infty}\Pr(|n\widehat V(A;Z) - n\,\mathrm{var}(\hat\beta)| > \epsilon) = 0$, then confidence intervals formed as $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V(A;Z)}$ will have $100(1-\alpha)\%$ coverage for $\beta$ in large $n$." "Lemma 3 follows directly from Lemma 2 and Slutsky's Theorem." [As printed, pp. 2247–2248] "Proposition 3. Given Assumption 2, if a variance estimator $\widehat V(A;Z)$ satisfies $\lim_{n\to\infty}\Pr(n\,\mathrm{var}(\hat\beta) - n\widehat V(A;Z) > \epsilon) = 0$, then confidence intervals formed as $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V(A;Z)}$ will have at least $100(1-\alpha)\%$ coverage for $\beta$ in large $n$." [Hypotheses attached] The setting of Section 2 and Assumption 2; $\widehat V(A;Z)$ is any real-valued statistic; the premise is for every $\epsilon > 0$. Read as: (Lemma 3) $\lim_{n\to\infty}\Pr\left(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\max\{\widehat V, 0\}}\right) = 1 - \alpha$, and (Proposition 3) the limit inferior of the same probability is at least $1 - \alpha$, for every $0 < \alpha < 1$.

: SAP.P011c.T39WholeStatement

Theorem: SAP.P011c.T39

Statement hash: 88255b232ec18b954f902751d9ca58099fc6e358fd0a873e14363c698f8159df

Statement definition: SAP.P011c.T39WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:36:47Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.conditional_wald_coverage: observed not used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T40

Corollary 4

p. 2248 (PDF p. 11)

Corollary 4, p. 2248. Under Assumptions 2 and 3 and valid constraints, both the maximizing homoskedastic estimator and the degree-only estimator are everywhere nonnegative, and their printed intervals have at least nominal asymptotic coverage at every level strictly between zero and one. Their values are measurable even for an arbitrary maximizing selector. Supplied degree caps need not be uniformly bounded; weighted means may differ. No relation between successive observed graphs or monotonicity of sample sizes is required.

Historical component credits

Proved AronowAudit.P011.corollary_four Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': 'P011/P011_CorollariesCompletion.lean:96 gives both liminf coverage bounds in T40 under scaling, homoskedasticity, valid constraints and samplewise V2 maximization. Although ConservativeCoverage uses sqrt(max(V,0)), V2 and V2prime are nonnegative for every observation by their formulas, as stated explicitly at P011/P011_Audit.lean:1905. Their clipped and printed coverage events are therefore identical, and no extra positive-variance, mean-spread or selector-measurability assumption is added.'}
Proved AronowAudit.P011.homoskedastic_estimators_nonnegative Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The word homoskedastic in the theorem name identifies the estimators, not a hidden assumption. The deterministic sum-of-squares formulas already imply nonnegativity, including their stated finite-vector domain.
Proved AronowAudit.P011.sample_variance_nonnegative Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Used inside the estimator nonnegativity declaration; no separate invocation is needed.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:da2c389252b2520dcbfb55620fe22bbd6968d785b916b3cff5d42e96612c49fb

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): [As printed, p. 2248] "Corollary 4. Given Assumptions 2 and 3, confidence intervals formed as $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V_2(A^m;Z)}$ or $\hat\beta \pm z_{1-\alpha/2}\sqrt{\widehat V_2'(Z)}$ have at least $100(1-\alpha)\%$ coverage for $\beta$ in large $n$." "Proofs for Corollaries 3 and 4 follow directly from Corollaries 1 and 2 and Proposition 3." [Hypotheses attached] The setting of Section 2; $A^m \in \mathcal A_2^m$; valid constraints; Assumptions 2 and 3. Read as: for every $0 < \alpha < 1$, $\liminf_{n\to\infty}\Pr\left(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V}\right) \ge 1 - \alpha$ for $\widehat V = \widehat V_2(A^m;Z)$ and for $\widehat V = \widehat V_2'(Z)$.

: SAP.P011c.T40WholeStatement

Theorem: SAP.P011c.T40

Statement hash: 00900493d7065929ab1a80b727f71bbe208a4742b537ee90f1b61d30b42c36b4

Statement definition: SAP.P011c.T40WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:36:47Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P011.corollary_four: observed not used

AronowAudit.P011.sample_variance_nonnegative: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P011-T41

Finite capacity-matrix and fixed-correction components of the broader condition-(C) repair accepted in the 19 October review

Accepted repair of P011-E001, runs/P011/writeup/rev01/draft/P011-rev01.tex, broader sufficient condition and fixed corrections after maximization

accepted repair of P011-E001, broader sufficient condition and fixed corrections after maximization in the 19 October review draft. The capacity matrix always equals the signless Laplacian plus the displayed real diagonal correction. The capacity condition makes it positive semidefinite and, for positive sample size, makes every normalized quadratic form nonnegative. Adding a graph-independent constant preserves all maximizers over the same compatible graphs, including ties. The decomposition, positivity, and maximizer equivalence include size zero; only division by sample size requires positivity.

Historical component credits

Proved AronowAudit.P011.recruitmentCapacityMatrix_psd Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This row covers exactly the sufficient finite PSD component of T41. PSD expands to symmetry and nonnegativity for every real vector; natural capacities and real subtraction match the target. Other T41 components are assessed separately in targets.json.
Proved AronowAudit.P011.maximizes_add_constant_iff Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This row covers exactly the fixed-correction maximizer component of T41. Neither the target nor the row claims the full asymptotic condition-(C) theorem.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:721343d25d5e36fdefac070401eaec6307cbdbdee0422d2c20bbf842b8a4858a

Standing assumptions: The article is Electronic Journal of Statistics 12 (2018) 2238–2252. Printed page = PDF page + 2237 (printed p. 2240 is PDF p. 3). Everything quoted from the article was read from the page images (staged/pages/p-NN.png, re-rendered at 200 dpi for the formulas). (S1) Graph and dependency graph. p. 2240 (PDF p. 3): "Consider a simple graph $\mathcal G = (\mathcal V, \mathcal E)$ with no parallel edges or self-loops. Let $|\mathcal V| = N$. Associated with each vertex $i \in \mathcal V$ is a random variable $Y_i$, and $\mathcal G$ characterizes probabilistic dependencies in the outcomes [e.g., 3]." "Definition 1 (Dependency graph). $\mathcal G$ is a dependency graph if for all disjoint sets $\mathcal V_1, \mathcal V_2 \subset \mathcal V$ with no edge in $\mathcal E$ connecting a vertex in $\mathcal V_1$ to a vertex in $\mathcal V_2$, the set $\{Y_i : i \in \mathcal V_1\}$ is independent from the set $\{Y_j : j \in \mathcal V_2\}$." (S2) Observation. p. 2240 (PDF p. 3): "Suppose $\mathcal G$ is a dependency graph and we observe a subset $\mathcal V_S \subseteq \mathcal V$, where $|\mathcal V_S| = n$. Label these observed vertices $1, \ldots, n$, and label the unobserved vertices in $\mathcal V \setminus \mathcal V_S$ arbitrarily by $n+1, \ldots, N$. For each $i \in \mathcal V_S$, we observe the outcomes $Y_1, \ldots, Y_n$, a fixed vector of coefficients $\theta = \theta_1, \ldots, \theta_n$, and the degrees $d_i = |\{j : \{i,j\} \in \mathcal E\}|$ for each $i \in \mathcal V_S$." "Definition 2 (Induced subgraph). For a set of vertices $\mathcal V_S \subseteq \mathcal V$, the induced subgraph in $\mathcal G$ is $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$, where $\mathcal E_S = \{\{i,j\} : i \in \mathcal V_S,\ j \in \mathcal V_S,\ \text{and } \{i,j\} \in \mathcal E\}$." "Let $\mathcal G_S = (\mathcal V_S, \mathcal E_S)$ be the induced subgraph of the observed vertices $\mathcal V_S$. It follows that $\mathcal G_S$ is also a dependency graph. Let $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$ be a subgraph of $\mathcal G_S$, consisting of all the observed vertices in $\mathcal V_S$, and a subset of the edges in $\mathcal E_S$." "Assumption 1 (Observed data). We observe the random outcomes $Y_1, \ldots, Y_n$, the fixed degrees $d_1, \ldots, d_n$, the fixed coefficients $\theta_1, \ldots, \theta_n$, and the fixed recruitment graph $\mathcal G_R$." "Let $Y = (Y_1, \ldots, Y_n)$, $d = (d_1, \ldots, d_n)$, and denote the observed data as $Z = (Y, d, \theta, \mathcal G_R)$." (S3) Estimator and target. p. 2240 (PDF p. 3): "Definition 3 (Linear unbiased estimator (LUE)). An estimator $\hat\beta$ is a linear unbiased estimator (LUE) of $\beta$ if $\hat\beta = \frac{1}{n}\sum_{i \in \mathcal V_S} \theta_i Y_i$ where $\mathrm{E}[\hat\beta] = \beta$." p. 2241 (PDF p. 4): "In what follows, we wish to conduct inference on $\beta$ given $Z$. We proceed by constructing conservative estimators of $\mathrm{var}(\hat\beta) = \frac{1}{n^2}\sum_{i \in \mathcal V_S}^{n}\sum_{j \in \mathcal V_S}^{n} \theta_i\theta_j\,\mathrm{cov}(Y_i, Y_j)$." (S4) Compatible adjacency matrices. p. 2241 (PDF p. 4): "We consider a class of variance estimators that depend on knowledge of $\mathcal G_S$, whose structure is represented by an $n \times n$ binary symmetric zero-diagonal adjacency matrix in which rows and columns are ordered by the indices $1, \ldots, n$ of the vertices in $\mathcal V_S$." "Definition 4 (Compatibility). The $n \times n$ binary symmetric adjacency matrix $A$ is compatible with the observed data $Z$ if for each $\{i,j\} \in \mathcal E_R$, $A_{ij} = A_{ji} = 1$, and for each $i \in \mathcal V_S$, $\sum_{j \in \mathcal V_S} A_{ij} \le d_i$." "The last condition in Definition 4 requires that the degree of $i$ in the subgraph $\mathcal G_S$ not be greater than its degree in the full graph $\mathcal G$. Let $A^{\mathcal O} = \{A^{\mathcal O}_{ij}\}$ be the true $n \times n$ adjacency matrix of $\mathcal G_S$, where $A^{\mathcal O}_{ij} = 1$ if $\{i,j\} \in \mathcal E_S$ for $i, j \in \mathcal V_S$ and 0 otherwise. Let $\mathcal A(Z) = \{A : A \text{ is compatible with } Z\}$ in the sense of Definition 4; it is clear that $A^{\mathcal O} \in \mathcal A(Z)$." "Definition 5 (Oracle estimator). For a family of variance estimators $\widehat V(A;Z)$ defined for $A \in \mathcal A(Z)$, the oracle estimator is $\widehat V(A^{\mathcal O};Z)$." "For a variance estimator $\widehat V(A;Z)$, define the set $\mathcal A^m = \{A \in \mathcal A(Z) : \widehat V(A;Z) \text{ is maximized}\}$." "Definition 6 (Maximal compatible estimator). Let $A^m \in \mathcal A^m$. The maximal compatible estimator is $\widehat V(A^m;Z)$." (S5) Asymptotic scaling. p. 2242 (PDF p. 5): "Assumption 2 (Asymptotic scaling). Consider the sequence $(\mathcal G, Z)_n$ of nested graphs $\mathcal G$ and observed data $Z = (\mathcal G_R, Y, d)$, where $\mathcal G_R = (\mathcal V_S, \mathcal E_R)$, $|\mathcal V_S| = n$, and $|\mathcal V| = N_n \ge n$. Assume there exist finite, positive constants $c_1$, $c_2$ such that for every element $(\mathcal G, Z)_n$, $\Pr(|\theta_i Y_i - \beta| > c_1) = 0, \forall i \in \mathcal V_S$ (bounded outcome values) and $\sum_{j \in \mathcal V_S} A^{\mathcal O}_{ij} \le c_2, \forall i \in \mathcal V_S$ (bounded degrees in dependency graph). Further assume there exists a finite, positive constant $c_3$ such that $\lim_{n\to\infty} n\,\mathrm{var}(\hat\beta) = c_3$ (nondegenerate limiting variance). Finally, assume $c_4$ is a finite positive constant such that $|\theta_i| < c_4$." (S6) General estimator. p. 2242 (PDF p. 5): "Define the plug-in sample variance $\hat\sigma_\theta^2 = n^{-1}\sum_{i \in \mathcal V_S}(\theta_i Y_i - \hat\beta)^2$, and the estimator $\widehat V_1(A;Z) = \frac{1}{n^2}\left[n\hat\sigma_\theta^2 + \sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}(\theta_i Y_i - \hat\beta)(\theta_j Y_j - \hat\beta)\right]$. (3.1)" p. 2244 (PDF p. 7): "Let $\mathcal A_1^m = \{A \in \mathcal A(Z) : \widehat V_1(A;Z) \text{ is maximized}\}$ be the set of compatible adjacency matrices that maximize $\widehat V_1(A;Z)$. Let $\Theta = \mathrm{diag}(\theta)$ be the $n \times n$ matrix with $\Theta_{ii} = \theta_i$ and $\Theta_{ij} = 0$ for $i \ne j$." (S7) Homoskedastic estimators. p. 2246 (PDF p. 9): "Assumption 3 (Homoskedasticity). $\mathrm{var}(\theta_i Y_i) = \mathrm{var}(\theta_j Y_j), \forall i, j \in \mathcal V$." "A bound that is relatively computationally simple to compute can be derived by noting that when $\mathrm{var}(\theta_i Y_i) = \sigma_\theta^2$, $\mathrm{cov}(\theta_i Y_i, \theta_j Y_j) \le \sigma_\theta^2 A^{\mathcal O}_{ij}$. Define the estimator $\widehat V_2(A;Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S}\sum_{j \in \mathcal V_S} A_{ij}\right]$. (3.7)" "Define $\mathcal A_2^m = \{A \in \mathcal A(Z) : \widehat V_2(A;Z) \text{ is maximized}\}$, and let $A^m \in \mathcal A_2^m$." pp. 2246–2247 (PDF pp. 9–10): "For $i \in \mathcal V_S$, let $d_i' = \min\{d_i, n-1\}$ be the degree of $i$ in $\mathcal G$, truncated at $n-1$. Let $\widehat V_2'(Z) = \frac{\hat\sigma_\theta^2}{n}\left[1 + \frac{1}{n}\sum_{i \in \mathcal V_S} d_i'\right]$. (3.9)" (S8) Degree bounds in place of degrees. p. 2248 (PDF p. 11): "Alternatively, we may not have full knowledge of the degrees $d = (d_1, \ldots, d_n)$, and instead have only an upper bound $d_i^*$ for each $d_i$, or a global upper bound $d_i \le d^*$ for all $i = 1, \ldots, n$. Conservative variance estimation in both of these cases can be achieved (by substituting $d_i^*$ or $d^*$ for $d_i$) with no change to the programs (3.5) and (3.8) or to the asymptotic results given here. When no information about $\mathcal G_R$ or the degrees $d$ is available, setting every $d_i = d^* = n-1$ delivers a maximally conservative upper bound." (S9) Audit notation, not in the paper (from the validators' derivations and the from-scratch check of the correction). $X_i = \theta_i Y_i$; $\mu_i = \mathrm{E}[X_i]$; $\delta_i = \mu_i - \beta$ (so $\sum_i \delta_i = 0$ by Definition 3, but no $\delta_i$ need be 0); $u_i = X_i - \mu_i$; $e_i = X_i - \hat\beta$; $C$ is the $n \times n$ covariance matrix of $(X_1, \ldots, X_n)$; $S = I + A^{\mathcal O}$; $b_n = \frac{1}{n}\delta^{\mathsf T}(I + A^{\mathcal O})\delta = \frac{1}{n}\left[\sum_i \delta_i^2 + \sum_i\sum_j A^{\mathcal O}_{ij}\delta_i\delta_j\right]$; for a symmetric $n \times n$ matrix $M$, $\widehat V_M = \frac{1}{n^2}e^{\mathsf T}Me$, so that $\widehat V_1(A;Z) = \widehat V_{I+A}$; for an adjacency matrix $A$, $d_i^A = \sum_j A_{ij}$ and $D_A = \mathrm{diag}(d_i^A)$; $D' = \mathrm{diag}(d_i')$. By Definition 1, $C_{ij} = 0$ whenever $i \ne j$ and $A^{\mathcal O}_{ij} = 0$, so $n\,\mathrm{var}(\hat\beta) = \frac{1}{n}\mathrm{tr}\{(I + A^{\mathcal O})C\}$. (S10) Reading conventions (for the formalizer; conventions, not quotations). (a) At each $n$ there is one probability space carrying the outcomes; $\mathrm{E}$, $\mathrm{var}$, $\mathrm{cov}$, $\Pr$ refer to their joint law. The graphs, the degrees $d_i$ and the coefficients $\theta_i$ are non-random; they and $\beta = \mathrm{E}[\hat\beta]$ may depend on $n$ (the regression weights of Section 2 do). (b) The constants $c_1, \ldots, c_4$ of Assumption 2 do not depend on $n$, and its conditions hold for every $n$ and every $i \in \mathcal V_S$. (c) "$\lim_{n\to\infty}\Pr(\cdot > \epsilon) = 0$ for any $\epsilon > 0$" is a statement about the sequence indexed by $n$. (d) "will have (at least) $100(1-\alpha)\%$ coverage for $\beta$ in large $n$" is read as: the limit (limit inferior) of $\Pr(|\hat\beta - \beta| \le z_{1-\alpha/2}\sqrt{\widehat V})$ equals (is at least) $1 - \alpha$, for $0 < \alpha < 1$, with $z_{1-\alpha/2} = \Phi^{-1}(1 - \alpha/2)$. The printed interval is undefined when $\widehat V < 0$, which can happen for $\widehat V_1$; the corrected statements use $\max\{\widehat V, 0\}$. (e) Double sums over $i, j \in \mathcal V_S$ are over ordered pairs. (f) In (3.5), $\Theta Y - \hat\beta$ is the vector with entries $\theta_i Y_i - \hat\beta$, $1$ is the vector of ones, and $\preceq$, $\succeq$ are element-wise $\le$, $\ge$ (the print says "less-than"; P011-N001). (g) Assumption 3 is used only for $i, j \in \mathcal V_S$ (P011-N001); the common variance, which may depend on $n$, is written $\sigma_\theta^2$ in the paper and $\sigma_n^2$ in the corrected statements. (h) "Valid constraints" means $\mathcal E_R \subseteq \mathcal E_S$ and $d_i \ge \sum_j A^{\mathcal O}_{ij}$ for every $i \in \mathcal V_S$ (true degrees in $\mathcal G$, or upper bounds on them as in (S8)), which gives $A^{\mathcal O} \in \mathcal A(Z)$. (i) In Corollaries 1 and 3, $A^m \in \mathcal A_1^m$; in Lemma 1 and Corollaries 2 and 4, $A^m \in \mathcal A_2^m$. Scope of the list. Targets come from two sources. (1) Headline claims P011-H1 to P011-H7 are mathematical and are targets (H1 and H4 are false as printed and appear as results touched by P011-E001; H2, H3, H5, H6 and the formal part of H7 appear as headline results). P011-H8 (the R package implements the method) is a statement about software and is not a target. (2) Confirmed errors whose correction changes a mathematical statement, a formula or a proof step: P011-E001 (Critical; targets T1–T10, T31–T33, with the corrected statements taken from the from-scratch check runs/P011/correction-recheck/recheck.json, as NOTES-FOR-AGENTS.md directs), P011-E004 (Moderate; T11, T12, T34), P011-E002 (Moderate; T13, T14, T30), P011-N006 (Moderate; an interpretive finding about the package vignette and README that rests on an exact counterexample; T15, T16, T35), and the Minor errors P011-E003 (T17, T18), P011-E005 (T19, T20), P011-E006 (T21), P011-E007 (T22, T23), P011-N001 (T24) and P011-E015 (T25, T26; the package's variance helpers use divisor $n-1$ where the paper's formulas (3.1), (3.7), (3.9) use $n$, a wrong constant relative to the printed formulas). Not targets: P011-E009 and P011-N007 (misprinted words), P011-N002 (bibliography), P011-E011, P011-E013 and P011-E014 (the package stops with an error or does not check the solver status: no formula or statement to correct), P011-E012 (the help page and vignette name estimator (3.9) where the code returns (3.7) at the optimized graph; both formulas and Lemma 1, which relates them, are correct as printed and are target T28, so there is no mathematical statement to correct), P011-E010 (out of scope), P011-E008 (clarity note), P011-N003 and P011-N008 (undetermined), P011-N004 and P011-N005 (not confirmed).

Effective claim (source TeX): Let R be a finite simple undirected known-edge graph, r_i its degrees, and h_i natural-number caps. Set K_R=I+A_R+diag(h_i-r_i), with subtraction over the reals. Then K_R=(A_R+diag(r_i))+diag(1+h_i-2r_i). If h_i≥2r_i−1 for every i, K_R is positive semidefinite and δᵀK_Rδ/n≥0 for every real vector δ. For any graph objective f and any graph-independent real correction c, the maximizers of f+c over exactly the same compatible graphs are exactly the maximizers of f, including ties.

: SAP.P011c.T41WholeStatement

Theorem: SAP.P011c.T41

Statement hash: f2822a2b7f43d7caa91a4984d05b63a0f04d2c4e09c0cbf41259418f5ac23b4c

Statement definition: SAP.P011c.T41WholeStatement

Certificate. 20261104T181331Z

Declaration review: faithful; 2026-11-04T18:36:47Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

Summary

What was and was not checked

Headline claims and related corrections

Corrections

Critical (1)

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

My comment and the history of this entry

Moderate (2)

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

Minor (14)

As printed

The problem

Evidence a derivation

What survives

Corrected

As printed

The problem

Evidence a derivation

What survives

Corrected

As printed

The problem

Evidence a derivation

What survives

Corrected

Related works and software

History of this entry

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem

Evidence a check of the printed page

What survives

Corrected

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem

Evidence an exact counterexample

What survives

Corrected

Related works and software

As printed

The problem