Rejoinder P047

Corrections to: Rejoinder: Nonparametric Identification Is Not Enough, but Randomized Controlled Trials Are

P. M. Aronow, James M. Robins, Theo Saarinen, Fredrik Sävje, and Jasjeet Sekhon (2025). Rejoinder: Nonparametric Identification Is Not Enough, but Randomized Controlled Trials Are. Observational Studies 11(1):85–90.

DOI: 10.1353/obs.2025.a956844
Audited edition: Observational Studies, 2025, 11(1):85–90; 7 pages in the audited file
SHA-256 of the audited file: 8a047af484c5141166876cd81ad4250f8be85fbd963c7810940466025bf700fc
Permanent address: https://pmaronow.github.io/llm-audit/works/P047/

Summary

The rejoinder restates the two points of the target article and replies to six commentaries. With independent and identically distributed sampling, ignorability, and fixed known common bounds on outcomes and positivity, a known propensity score permits uniformly root- $n$ consistent estimators and finite-sample valid confidence intervals. With an unknown, unrestricted propensity score and unrestricted outcome regressions over continuous covariates, no uniformly consistent estimator exists without further restrictions. The rejoinder has no proofs, tables or data. 4 entries are classified as errors (4 Minor), and 1 is a clarity note; the errors concern a statement about FDA rules, a pair of word-form errors, an address and a licence line. Three possible errors were not confirmed.

What was and was not checked

I checked the following points, with the qualifications stated below.

Section 1, first bullet (p. 85), for independent and identically distributed units with conditionally ignorable treatment, a known propensity score bounded away from 0 and 1 by a fixed known common constant, and a fixed known common bound on outcomes. A derivation shows that the Horvitz–Thompson estimator is unbiased and uniformly root- $n$ consistent, that a Hoeffding interval is valid at every $n$ with width proportional to $n^{-1/2}$ , and that the estimator is asymptotically normal at each fixed law, possibly with zero limiting variance. The rejoinder does not restate the bound; the preprint of the target article defines positivity using a known positive constant. I used bounds fixed and common to the model for the uniform guarantee.
Section 1, second bullet (pp. 85–86). Suppose that the propensity score and the outcome regression are unknown and unrestricted, and that a covariate is continuous. Two separate constructions show that no estimator is then uniformly consistent and that no honest interval has shrinking width, even for binary outcomes and a propensity score bounded away from 0 and 1. The word “untestable” in this bullet is the subject of P047-N003.
Section 2.4 (p. 87). For ignorable assignment conditional on covariates, a direct derivation shows that the assignment mechanism drops out of the outcome-parameter posterior under distinct parameters and prior independence, the two conditions that the rejoinder names.
Section 2.5 (p. 88), the rate $n^{1/4}$ . The exponent is positive on the page and in the manuscript source, in the convention of “root- $n$ ” on p. 85. Assuming the usual bounded-regression risk balance and uniform entropy exponent $2V/(V+2)$ for a finite $V$ , a calculation gives the rate exponent $1/4 + 1/(4(V+1))$ , which exceeds $1/4$ .
Section 2.5 (p. 88), Wald-type intervals centred on a superefficient estimator. An exact calculation with the Hodges estimator of a normal mean illustrates the failure of uniform coverage that the sentence describes.
The numbers in the text ( $n^{1/4}$ and the sample sizes 100,000 and 200 on p. 88, and 0.05 on p. 89), which agree with the manuscript source.
The single reference, Robins and Ritov (1997), against the bibliography file of the manuscript source: authors, year, title, journal, volume, issue and pages agree.
The published text against the manuscript source. The bodies agree word for word, and the title was lengthened in production from “Rejoinder”.
Internal references and consistency: the section numbers run consecutively, the single footnote and its mark are on p. 88, the single citation has its entry in the reference list, and the replies of Sections 2.1, 2.5 and 2.6 are consistent with the two points of Section 1.

I did not check the following.

The target article, apart from its positivity assumption and its results for a known propensity score, which were read in the preprint. The recapitulation in Section 1 was not otherwise compared with it; this includes any definition of “untestable”, the quotation from Judea Pearl, and the statement on p. 88 that the target article never claimed RCTs to be universally superior to observational studies.
The six commentaries, which are not in the reference list. I did not compare with them the paraphrases, the three direct quotations (pp. 88 and 89), the pointers to “their Section 7” and “their Section 8”, or the sample sizes 100,000 and 200.
Robins and Ritov (1997), which was not read. The attribution of the two results of Section 1 to it was not checked.
The minimax theorem behind the rate $n^{1/4}$ , its loss and the entropy bound for the class, and the superefficient estimators of the commentary’s Section 8.
Which licence governs the rejoinder (see P047-N001).

The rejoinder has no proofs, numbered results, tables, figures, appendix, data or replication code. These checks do not show that the rejoinder is correct in the respects that were not examined.

Headline claims and related corrections

For averages of bounded potential outcomes under ignorability and positivity given at least one continuous covariate: if the propensity score is known, as in an RCT, there are simple estimators that are uniformly root- $n$ consistent and asymptotically normal, and confidence intervals based on them are finite-sample valid with widths shrinking at a root- $n$ rate.
Section 1, first bullet, p. 85 (PDF p. 2); no numbered result; restates the target article and Robins and Ritov (1997)
If the propensity score is not known, as in an observational study, there is no uniformly consistent estimator and no uniform (honest) large-sample confidence interval whose width shrinks with the sample size; to obtain these the practitioner must impose untestable assumptions on either the propensity score function or the conditional expectation function of the outcomes.
Section 1, second bullet, pp. 85-86 (PDF pp. 2-3); no numbered result; attributed to Robins and Ritov (1997)
See P047-N003.
The known-propensity and unknown-propensity results are attributed to Robins and Ritov (1997). With sample size and identifying assumptions held equal, an RCT has an estimation advantage because its propensity score need not be estimated; comparing RCTs and observational studies only by the credibility of ignorability and positivity overlooks this advantage.
Section 1, last paragraph, p. 86 (PDF p. 3); restated in Section 2.5, p. 88 (PDF p. 5); the two bullets of Section 1
For the class of multivariate cadlag propensity score functions with bounded sectional variation norm discussed by Phillips and van der Laan, the minimax rate exceeds $n^{1/4}$ in normalizing-rate notation for each fixed finite covariate dimension. The practical plausibility of this restriction requires justification beyond describing the class as fairly large.
Section 2.5, third paragraph, p. 88 (PDF p. 5); the inline formula $n^{1/4}$ , the only formula in the work
See P047-E002.
The superefficient estimators described in Phillips and van der Laan's Section 8 pay a price: Wald-type intervals centred on them are no longer uniform asymptotic confidence intervals, even when the propensity score and outcome regressions are cadlag with bounded sectional variation norm.
Section 2.5, last paragraph, p. 88 (PDF p. 5); no numbered result
An observational study can be preferable to an RCT, including for ethical or financial reasons. Phillips and van der Laan's example with 100,000 observational subjects and 200 randomized subjects addresses a claim of universal RCT superiority; the target article makes its comparison with sample size and identifying assumptions held equal.
Section 2.5, first and second paragraphs, pp. 87-88 (PDF pp. 4-5); a paraphrase of the comment and of the target article
The title's 'enough' concerns estimation given identification. RCTs do not by themselves ensure generalizability or external validity or solve every methodological challenge. The complexity of naturally occurring propensity scores and how often the restrictions proposed by Phillips and van der Laan hold in practice are empirical questions; the target article supplies no evidence on that complexity.
Section 2 opening and Section 2.1, p. 86 (PDF p. 3); Section 2.5, fourth paragraph and footnote 1, p. 88 (PDF p. 5)
The central purpose of statistical methods is to produce good evidence about the world, including evidence for drug approval. Clear but incomplete rules for regulation cannot replace scientific integrity and methodological sophistication.
Section 2.6, pp. 89-90 (PDF pp. 6-7); no numbered result
See P047-E001.

Corrections

Minor (4)

Real but inconsequential, for example a misprint that does not change a result.

P047-E001 Minor

The FDA does not require evidence from RCTs for every drug approval; it ordinarily expects randomized evidence of efficacy and allows other adequate and well-controlled designs.

Section 2.6, third paragraph, p. 89 (PDF p. 6), sentence beginning “The reason the U.S. Food and Drug Administration”

As printed

The reason the U.S. Food and Drug Administration requires evidence from RCTs for drug approval is not because it successfully adjudicates the question at hand, but because such evidence is effective in producing desirable policy outcomes.

The problem

The sentence says without qualification that the FDA requires evidence from RCTs for drug approval. Read as a rule for every approval, this is false. The statute asks for “adequate and well-controlled investigations”, and the regulation that defines such studies lists a historical control among the accepted designs. The sentence does not say “every approval”, and it can be read as a description of what the FDA ordinarily expects, which the sources do not contradict. The unqualified verb overstates the rule.

Evidence a check of the printed page

I checked public sources in editions that precede the rejoinder. The statute, 21 U.S.C. 355(d), defines substantial evidence as “evidence consisting of adequate and well-controlled investigations, including clinical investigations”. The regulation, 21 CFR 314.126, lists “Historical control” in paragraph (b)(2)(v) and says in paragraph (b)(4) that “Ordinarily, in a concurrently controlled study, assignment is by randomization, with or without stratification.” The FDA approved Nulibry (fosdenopterin) on 26 February 2021 on a comparison of 13 treated patients with 18 matched, untreated patients, and its epidemiology review describes the population of the efficacy study as “Nulibry patients from 3 single-arm trials and untreated natural history controls”. The same application contained a randomized, blinded, placebo-controlled phase 1 study in healthy volunteers. The approval shows efficacy established without randomization; it is not an example of an application with no RCT evidence.

What survives

Only this illustration uses the statement. The distinction between producing evidence and merely reaching a decision, and the comparison of RCTs with tea leaf readings, do not depend on a requirement without exceptions. The correction weakens the description of the FDA’s rule, and it does not address the reason that the sentence gives for the FDA’s practice.

Corrected

The reason the U.S. Food and Drug Administration ordinarily expects randomized evidence of efficacy for drug approval, while allowing other adequate and well-controlled designs, is not because it successfully adjudicates the question at hand, but because such evidence is effective in producing desirable policy outcomes.

A wording that makes no statement about the FDA’s rule also keeps the argument: “The value of evidence from RCTs in drug approval lies not merely in adjudicating the question at hand, but in helping produce desirable policy outcomes.”

Crop of five lines of Section 2.6. The sentence at issue reads: The reason the U.S. Food and Drug Administration requires evidence from RCTs for drug approval is not because it successfully adjudicates the question at hand, but because such evidence is effective in producing desirable policy outcomes. A red outline surrounds the words requires evidence from RCTs for drug approval. The next sentence reads: RCTs are preferable over tea leaf readings because of their evidence-producing qualities. — Section 2.6, third paragraph, p. 89 (PDF p. 6). The outline marks the statement that the FDA requires evidence from RCTs for drug approval.

P047-E002 Minor

On p. 87, “where upon” should read “whereupon”; on p. 88, “a cursorily declaration” should read “a cursory declaration”.

Section 2.4, p. 87 (PDF p. 4), sentence beginning “They highlight that these assumptions”; Section 2.5, third paragraph, p. 88 (PDF p. 5), sentence beginning “They claim that these assumptions are realistic”

As printed

p. 87. They highlight that these assumptions do not hold in many settings, where upon the assignment mechanism becomes relevant also to a Bayesian.

p. 88. They claim that these assumptions are realistic, but they provide no justification for this position other than a cursorily declaration that the function class is fairly large.

The problem

On p. 87 the conjunction “whereupon” is split into two words. On p. 88 the adverb “cursorily” stands where the adjective “cursory” is needed to modify “declaration”.

Evidence a check of the printed page

I checked the two pages. Both forms also occur in the manuscript source.

What survives

The meaning of both sentences is unchanged, and no statement of the rejoinder depends on either word.

Corrected

p. 87. They highlight that these assumptions do not hold in many settings, whereupon the assignment mechanism becomes relevant also to a Bayesian.

p. 88. They claim that these assumptions are realistic, but they provide no justification for this position other than a cursory declaration that the function class is fairly large.

Crop of five lines of Section 2.4. The third line reads: settings, where upon the assignment mechanism becomes relevant also to a Bayesian. They. A red outline surrounds the two words where upon. — Section 2.4, p. 87 (PDF p. 4). The outline marks “where upon”.

Crop of six lines of Section 2.5. The sentence at issue reads: They claim that these assumptions are realistic, but they provide no justification for this position other than a cursorily declaration that the function class is fairly large. A red outline surrounds the word cursorily, at the start of the fourth line. — Section 2.5, third paragraph, p. 88 (PDF p. 5). The outline marks “cursorily”.

P047-N001 Minor

The cover page names the licence in words as Creative Commons Attribution 4.0, while the icon beside those words and the footer of p. 85 give CC BY-NC-ND 4.0.

Cover page added by the distributor (PDF p. 1), licence line at the foot; compare the copyright footer of p. 85 (PDF p. 2)

As printed

Cover page, beside an icon labelled BY NC ND. This work is licensed under a Creative Commons Attribution 4.0 International License.

Footer of p. 85. ©2025 P. M. Aronow, James M. Robins, Theo Saarinen, Fredrik Sävje and Jasjeet Sekhon. This is an open access article distributed under the terms of a Creative Commons license (CC BY-NC-ND 4.0).

The problem

The published PDF identifies its licence in three places, and the three do not agree. The words on the cover page name the Attribution licence. The icon on the same line and the footer of the first page of the rejoinder give a different licence, Attribution-NonCommercial-NoDerivatives. The cover page is a wrapper added by the distributor, and it is not part of the text of the rejoinder.

Evidence a check of the printed page

I compared the two pages. They establish the mismatch, and they do not establish which licence governs the rejoinder.

What survives

No statement of the rejoinder is affected. The correction below makes the words of the cover page agree with the icon beside them and with the footer of p. 85.

Corrected

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Crop of one line at the foot of the cover page. At the left is a Creative Commons icon with the labels BY, NC and ND. To its right, grey text reads: This work is licensed under a. A red outline surrounds the words that follow: Creative Commons Attribution 4.0 International License. — Foot of the cover page (PDF p. 1). The outline marks the licence as named in words, Creative Commons Attribution 4.0 International License. The icon at the left of the same line is labelled BY NC ND.

Crop of the two-line copyright footer of the first page of the rejoinder. It reads: Copyright 2025 P. M. Aronow, James M. Robins, Theo Saarinen, Fredrik Sävje and Jasjeet Sekhon. This is an open access article distributed under the terms of a Creative Commons license (CC BY-NC-ND 4.0). A red outline surrounds the words CC BY-NC-ND 4.0. — Copyright footer of p. 85 (PDF p. 2), for comparison. The outline marks the licence named there, CC BY-NC-ND 4.0.

P047-N002 Minor

The address of the Harvard School of Public Health should read Boston, MA 02115, not Cambridge, MA 02115.

Author block, address under “James M. Robins”, p. 85 (PDF p. 2)

As printed

James M. Robins

Harvard School of Public Health

Cambridge, MA 02115, USA

The problem

The city does not match the school and the ZIP code that are printed with it. The school gives its primary mailing address as 677 Huntington Avenue, Boston, MA 02115. The manuscript source has the same line as the published page.

Evidence a check of the printed page

I checked the page against two public sources. The contact page of the school, now the Harvard T.H. Chan School of Public Health, gives the primary mailing address above. An archived labelling list of the U.S. Postal Service pairs the ZIP code 02115 with the label “BOSTON MA 02115”.

What survives

Only the city changes. The correction makes no claim about a street address or an office, and no statement of the rejoinder is affected.

Corrected

James M. Robins

Harvard School of Public Health

Boston, MA 02115, USA

Clarity notes

Imprecise or ambiguous wording whose intended meaning is correct; not an error.

P047-N003 Clarity note

In the second bullet, “untestable” means that the data cannot verify the added assumptions; it does not mean that those assumptions have no refutable implications.

Section 1, second bullet, last sentence, pp. 85–86 (PDF pp. 2–3); the word is in the first line of p. 86

See also under Formal statements (Lean): P047-T3

As printed

To achieve these properties, the practitioner must impose untestable assumptions on either the propensity score function or the conditional expectation function of the outcomes.

Intended meaning

The bullet concerns an unknown propensity score $\pi(w)$ , the probability of treatment at the covariate value $w$ . The sentence says that uniform consistency and honest intervals of shrinking width then require further restrictions on $\pi$ or on the outcome regression, and that the data cannot certify such restrictions. Sections 2.3 and 2.5 support this reading, since they discuss how hard it is to judge whether a propensity score belongs to a given functional class. The sentence is correct on this reading. I take a uniform covariate and two laws that differ only in the propensity score, with $\pi = 1/2$ in the first, and $\pi(w) = 1/2 - \eta$ for $w < 1/2$ and $\pi(w) = 1/2 + \eta$ for $w \ge 1/2$ in the second. In samples of $n$ independent and identically distributed units, the total variation distance between the two laws is at most $n\eta$ . Letting $\eta \to 0$ , I find that a procedure that certifies a constant propensity score with probability at most $\alpha$ under every nonconstant one also certifies it with probability at most $\alpha$ under the first law.

The word is too strong if it is read as “having no refutable implications”. Under this sampling assumption and the bullet’s ignorability assumption, a propensity score equal to an unknown constant $p \in [1/5, 4/5]$ is a restriction under which both properties hold for outcomes in $[0,1]$ , and the data can refute it. For $n = 20$ , an exact one-sided Fisher test of association between treatment and the indicator of $w \ge 1/2$ has size at most $1/20$ for every $p$ and every covariate distribution. With a uniform covariate, its power against a propensity score that jumps from $1/5$ to $4/5$ at $w = 1/2$ is

\frac{72825613746176}{95367431640625} \approx 0.764 .

The example restricts the model, and the impossibility result of the bullet is unaffected.

Corrected

To achieve these properties, the practitioner must impose complexity-reducing assumptions on either the propensity score function or the conditional expectation function of the outcomes.

The replacement is the rejoinder’s own term on p. 87. The sentence may instead keep “untestable” and say that it means not verifiable uniformly from the observed data, which does not exclude refutable implications.

Crop of four lines at the top of page 86. The first two lines end the second bullet: the practitioner must impose untestable assumptions on either the propensity score function or the conditional expectation function of the outcomes. A red outline surrounds the word untestable. The next two lines begin a new paragraph: These insights are decades old, going back to Robins and Ritov (1997), but they are not yet common knowledge. — Top of p. 86 (PDF p. 3): the end of the second bullet of Section 1 and the two lines that follow it. The outline marks the word “untestable”.

Crop of the three lines of the second bullet at the foot of page 85. They read: If the propensity score is not known, such as in an observational study, there exist neither uniformly consistent estimators nor uniform (i.e., honest) large-sample confidence intervals whose widths are shrinking with the sample size. To achieve these properties, and the page ends there. A red outline surrounds the words To achieve these properties. — Foot of p. 85 (PDF p. 2): the second bullet of Section 1. The outline marks the opening words of the sentence at issue, which continues on p. 86.

Formal statements (Lean)

Formal results: 3 whole; 0 restricted; 0 refuted; 0 omitted.

Coverage concerns the identified claim under its stated assumptions. A checked proof establishes the formal statement; correspondence to the published text was reviewed by AI. Repository access is currently required to open the source links.

Each result links to its current formal statement and proof. “Whole” covers the identified claim under its assumptions; “restricted” covers the stated narrower scope; “refuted” gives a counterexample to the identified assertion; “omitted” has no current theorem. Any named published theorem used as a premise remains an assumption. Browse the formal results for this work.

P047-T1

Section 1 (Recapitulation), first bullet, second sentence: finite-sample-valid confidence intervals of root-n width when the propensity score is known (no numbered result)

p. 85 (PDF p. 2)

Section 1, first bullet, second sentence, p. 85 (PDF p. 2). The known-propensity estimators have the stated conditional and unconditional means, contribution bounds, Hoeffding tails, finite-sample interval coverage, and exact root-n ATE interval width, including the one-mean component under its weaker one-arm conditions.

Historical component credits

Proved AronowAudit.P047.known_propensity_finite_sample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The target formal reading explicitly allows arbitrary W and fixes common known B and epsilon. Compared with the printed continuous-covariate setting this is a harmless extension, not a restriction; the stated non-strict overlap implies inherited strict overlap with a smaller positive bound. This theorem covers (i)-(v), with (vi) explicitly assigned to another row/chunk. I retain the claim-stage faithful classification because these facts lose no case and the target text already supplies the numerical bounds.
Proved AronowAudit.P047.treated_mean_finite_sample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; T1(vi) expressly contains these one-sided outcome and overlap premises and this exact four-part conclusion. Arbitrary covariate laws include the printed continuous-covariate cases; CausalLaw's separate-arm all-tests condition matches the inherited separate-arm ignorability, and no control-outcome bound or upper-overlap premise is introduced. The saved claim-stage verdict remains faithful.
Proved AronowAudit.P047.measurable_covariate Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.covariateSigma_le Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_covariate_conditioning Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_arm Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_potential Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_observed Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_armPropensity_conditioning Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_armIPW Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.score_eq_armIPW Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_score Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.integrable_potential Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.armPropensity_ge Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.armIPW_abs_le Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.score_abs_le Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.integrable_arm_numerator Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.integrable_armIPW Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.conditional_armIPW Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.conditional_score_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.score_unbiased Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.sample_coordinate_hasLaw Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.sample_scores_independent Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.sample_score_identDistrib Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.estimator_unbiased Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.subgaussian_abs_tail Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.bounded_independent_average_tail Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.estimator_hoeffding Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.measurable_estimator Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.symmetric_interval_coverage Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.halfWidth_exponential Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.confidenceInterval_coverage Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.confidenceInterval_root_width Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.propensity_is_conditional_probability Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The selected finite-sample aggregate theorems supply the target components; this supporting result is not separately invoked.
Proved AronowAudit.P047.confidenceInterval_endpoint_width Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The numerical target assumptions establish that the endpoints are ordered, so no target-width case is lost by using endpoint subtraction as the auxiliary geometry fact.
Proved AronowAudit.P047.armIPW_eq_observed Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Consistency is built into observed; the credited row is the correctly described auxiliary observability identity. No substantive disagreement with the recorded mathematics; the extra note records the totalized-division convention.
Proved AronowAudit.P047.treatedEstimator_eq_observed_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The right side is the inherited target article's observed-data HT formula, although the rejoinder itself prints no estimator formula. At n=0 both sides use the harmless zero-division convention; target assertions use n>=1.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:274dcfa80485a913443947a8edc732313fcf1cce8d8a3ccf1ecbb8b426f8526c

Standing assumptions: The work is a six-page prose rejoinder (Observational Studies 11.1 (2025): 85-90). Journal pp. 85-90 are PDF pp. 2-7, so printed page = PDF page + 83; PDF p. 1 is a Project MUSE cover page. It contains no theorem, proposition, lemma, numbered or displayed result, proof, estimator formula, table or figure; its only formula is the inline $n^{1/4}$ on p. 88 (PDF p. 5); it has one footnote and one reference (Robins and Ritov 1997). All seven page images were read, and everything quoted from the rejoinder below was transcribed from them. (S1) Setting of the two recapitulated results, Section 1, p. 85 (PDF p. 2): 'When a practitioner aims to estimate averages of bounded potential outcomes (e.g., the average treatment effect on a binary outcome) in a setting where both ignorability and positivity are known to hold after adjusting for at least one continuous covariate, the following statements are true:' Two bullets follow; they are targets P047-T1 to P047-T3. (S2) Status of the two bullets. Section 1, p. 85 (PDF p. 2): 'We believe this broader perspective is important, and the comments serve as complements to the somewhat narrow points we made in our paper. However, we believe these narrow points are of great consequence, and we find it useful to briefly recapitulate them here.' Section 1, p. 86 (PDF p. 3): 'These insights are decades old, going back to Robins and Ritov (1997), but they are not yet common knowledge.' 'Our paper' is the target article, 'Nonparametric identification is not enough, but randomized controlled trials are', Observational Studies 11.1 (2025): 3-16 (work P046 in this audit). It is not among the staged sources of P047 and is not in the rejoinder's reference list. The rejoinder calls ignorability and positivity 'the identifying assumptions' (p. 86, PDF p. 3: 'While the identifying assumptions indeed often are more credible in an RCT, the position that this is the only important difference is misleading'). (S3) What the rejoinder does not define. It uses no symbols and defines none of: potential outcomes, ignorability, positivity, propensity score, estimator, 'uniformly root-$n$ consistent', 'asymptotically normal', 'finite-sample valid', 'uniformly consistent', 'uniform (i.e., honest)', width. The definitions the bullets inherit are those of the target article, transcribed here from its page images (runs/P046/staged/pages/p-04.png to p-06.png; in that article printed page = PDF page + 1). Target article, Section 2, p. 5 (its PDF p. 4): 'We assume that we have an i.i.d. sample $(W_i, Y_i, X_i)$ from some large population, where $Y_i$ and $X_i$ are binary, and $W_i$ is drawn from a continuous distribution (either known or unknown). As above, $Y_i$ is the outcome, $X_i$ is the treatment, and $W_i$ is the covariate.' 'For our discussion, we invoke the consistency assumption (c.f., the stable unit treatment value assumption), $Y_i = (1 - X_i)Y_i(0) + X_iY_i(1)$, and unconfoundedness, $Y_i(x) \perp\!\!\!\perp X_i \mid W_i, \quad \forall x \in \{0, 1\}$.' 'We define the propensity score function as $\pi(w) = \Pr(X_i = 1|W_i = w)$, and assume that positivity holds, meaning that $\delta < \pi(W_i) < 1 - \delta$ with probability one for some positive $\delta$. Without loss of generality, the parameter of interest is the average potential outcome under treatment: $\mu = \mathrm{E}[Y_i(1)]$.' Target article, p. 6 (its PDF p. 5): 'Proposition 1 Suppose we have a study for which consistency, unconfoundedness, and positivity hold. Without further assumptions, there exists no estimator of $\mu$ that is uniformly consistent.' 'The focus on uniform consistency allows the population distribution to depend on $n$.' 'in an RCT, the propensity score function, $\pi(w)$, is known to the researcher, since it is specified by the experimental design.' 'Proposition 2 Suppose the propensity scores are known. Then the Horvitz–Thompson estimator $\hat\mu = n^{-1}\sum_{i=1}^{n}[\pi(W_i)]^{-1}X_iY_i$ is unbiased, uniformly root-n consistent, and asymptotically normal for $\mu$.' Target article, p. 7 (its PDF p. 6): 'This confidence interval is finite-sample valid, meaning that its coverage is guaranteed to be greater or equal to its nominal level even in small samples.' The external check recorded in runs/P047/settle/settled.json (item P047-A003) read the same positivity assumption in the preprint, arXiv:2108.11342v2 ('$\delta < \pi(W_i) < 1 - \delta$' 'with probability one for some known positive $\delta$ bounded away from 0'). (S4) Formal setting used in the targets. This is the notation of the audit's validators; the rejoinder prints none of it. Units $i = 1, \ldots, n$ are i.i.d. copies of $(W, A, Y(0), Y(1))$: $W$ is the covariate, with a continuous distribution; $A \in \{0, 1\}$ is the treatment ($X_i$ in the target article); $Y(0), Y(1)$ are the potential outcomes; the observed outcome is $Y = AY(1) + (1 - A)Y(0)$ (consistency). Ignorability: $A \perp Y(a) \mid W$ for $a = 0, 1$ (the target article's unconfoundedness; the validators write the joint form $A \perp (Y(0), Y(1)) \mid W$, and their derivations use only $E[AY(1) \mid W] = e(W)\,E[Y(1) \mid W]$ and $E[(1 - A)Y(0) \mid W] = (1 - e(W))\,E[Y(0) \mid W]$). Propensity score: $e(w) = \Pr(A = 1 \mid W = w)$ ($\pi(w)$ in the target article). Bounded potential outcomes: $|Y(a)| \le B$. Positivity: $\varepsilon \le e(W) \le 1 - \varepsilon$ almost surely. $B$ and $\varepsilon$ are fixed constants common to the model. Estimands: $\tau = E[Y(1) - Y(0)]$ (the rejoinder's example, 'the average treatment effect') and $\theta = E[Y(1)]$ ($\mu$ in the target article). All probabilities and expectations are over repeated i.i.d. samples. $\log$ is the natural logarithm. 'Known' propensity score: the estimator and the interval may use the function $e$. 'Not known': they are functions of the observed sample $(W_i, A_i, Y_i)_{i \le n}$ alone, the same for every law in the model. (S5) What is not a target. No confirmed error changes a mathematical statement, a formula or a proof step. The four confirmed errors, all Minor (rate/ratings.json), are P047-E001 (the assertion about what the U.S. Food and Drug Administration requires, p. 89), P047-E002 (the word forms 'where upon', p. 87, and 'cursorily declaration', p. 88), P047-N001 (licence wording on the distributor's cover page, PDF p. 1) and P047-N002 (the city in an author address, p. 85). P047-A001, P047-A002 and P047-A003 are not confirmed and P047-N003 is a clarity note. There is therefore no corrected-statement, counterexample or as-printed target. Headline claims P047-H3, P047-H6, P047-H7 and P047-H8 are interpretive, or concern what the commentaries and the target article say. Two further headline claims are mathematical in character but cannot be stated from this work, and are not targets. P047-H4, Section 2.5, p. 88 (PDF p. 5): 'Namely, that it belongs to “the class of multivariate real-valued cadlag functions with a bound on the sectional variation norm.” This class is sufficiently small so that it is not subject to the curse of dimensionality. In fact, the minimax rate of convergence always exceeds $n^{1/4}$ regardless of the dimension of the covariates.' The function class is named only by a quotation from a commentary that is not staged; the rejoinder defines neither the class, the sectional variation norm, the quantity estimated, the loss nor the sampling model, attaches no hypothesis and gives no citation; and the validator did not verify the entropy or minimax premises (validated.json, P047-A001, what_is_not_established). P047-H5, Section 2.5, p. 88 (PDF p. 5): 'However, the price paid for superefficiency is that Wald-type intervals centered on the estimator are no longer uniform asymptotic confidence intervals, even when the propensity score and outcome regressions are cadlag with bounded sectional variation norm.' The estimators are described only in Section 8 of the same unstaged commentary (validated.json, P047-A002).

Effective claim (source TeX): [Printed: Section 1, p. 85 (PDF p. 2)] 'When a practitioner aims to estimate averages of bounded potential outcomes (e.g., the average treatment effect on a binary outcome) in a setting where both ignorability and positivity are known to hold after adjusting for at least one continuous covariate, the following statements are true: • If the propensity score is known, such as in a randomized controlled trial (RCT), there exist simple estimators that are uniformly root-$n$ consistent and asymptotically normal. Confidence intervals based on these estimators are finite-sample valid and their widths shrink at a root-$n$ rate.' The target is the second sentence of the bullet, with the exact finite-sample facts about the estimator on which it rests; the first sentence is P047-T2. [Formal reading. The rejoinder prints no formula. This is the statement that the audit's two validators derived (validator artifact, section 8; second validator, section 5), with positivity in the sense of the target article that the bullet recapitulates.] Let $(W_i, A_i, Y_i(0), Y_i(1))$, $i = 1, \ldots, n$, be i.i.d., with $A_i \in \{0, 1\}$, observed outcome $Y_i = A_iY_i(1) + (1 - A_i)Y_i(0)$, ignorability $A_i \perp Y_i(a) \mid W_i$ for $a = 0, 1$, and known propensity score $e(w) = \Pr(A_i = 1 \mid W_i = w)$. Let $0 < B < \infty$ and $0 < \varepsilon \le 1/2$ be fixed known constants with $|Y_i(a)| \le B$ for $a = 0, 1$ and $\varepsilon \le e(W_i) \le 1 - \varepsilon$ almost surely. Let $\tau = E[Y_i(1) - Y_i(0)]$, $Z_i = \frac{A_iY_i}{e(W_i)} - \frac{(1 - A_i)Y_i}{1 - e(W_i)}$ and $T_n = \frac{1}{n}\sum_{i=1}^{n} Z_i$. Then, for every $n \ge 1$: (i) $E[Z_i \mid W_i] = E[Y_i(1) - Y_i(0) \mid W_i]$, hence $E[Z_i] = \tau$ and $E[T_n] = \tau$; (ii) $|Z_i| \le B/\varepsilon$; (iii) for every $t \ge 0$, $\Pr(|T_n - \tau| \ge t) \le 2\exp\left(-\frac{n t^2 \varepsilon^2}{2B^2}\right)$; (iv) for every $0 < \alpha < 1$, with $h_n = \frac{B}{\varepsilon}\sqrt{\frac{2\log(2/\alpha)}{n}}$ and $C_n = [T_n - h_n,\ T_n + h_n]$, $\Pr(\tau \in C_n) \ge 1 - \alpha$; (v) the width of $C_n$ is the deterministic number $2h_n = \frac{2B}{\varepsilon}\sqrt{2\log(2/\alpha)}\; n^{-1/2}$, so $\sqrt{n}$ times the width does not depend on $n$. Statements (i)-(v) hold for every law that satisfies the conditions above with the same $B$ and $\varepsilon$; nothing is assumed about the distribution of $W_i$, the form of $e$, or the functions $w \mapsto E[Y_i(a) \mid W_i = w]$. (vi) [One potential-outcome mean, the form checked by one of the fresh readers.] Under the same i.i.d. sampling, consistency and ignorability, if $0 \le Y_i(1) \le 1$ and $e(W_i) \ge \varepsilon$ almost surely, then with $\theta = E[Y_i(1)]$ and $\hat\theta_n = \frac{1}{n}\sum_{i=1}^{n}\frac{A_iY_i}{e(W_i)}$: $E[\hat\theta_n] = \theta$; $0 \le \frac{A_iY_i}{e(W_i)} \le \frac{1}{\varepsilon}$; $\Pr(|\hat\theta_n - \theta| \ge t) \le 2\exp(-2n\varepsilon^2t^2)$ for every $t \ge 0$; and $\hat\theta_n \pm \frac{1}{\varepsilon}\sqrt{\frac{\log(2/\alpha)}{2n}}$ covers $\theta$ with probability at least $1 - \alpha$ for every $n \ge 1$ and every $0 < \alpha < 1$.

: T1Statement

Theorem: SAP.P047.T1

Statement hash: bdf8c71a29d919842eda9ecdcffae478298aa730086de4707ef1279c7999721e

Statement definition: SAP.P047.T1Statement

Certificate. 20261104T173313Z

Declaration review: faithful; 2026-11-04T17:56:00Z; 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.P047.armIPW_abs_le: observed used

AronowAudit.P047.armPropensity_ge: observed used

AronowAudit.P047.bounded_independent_average_tail: observed used

AronowAudit.P047.conditional_armIPW: observed used

AronowAudit.P047.conditional_score_identity: observed used

AronowAudit.P047.confidenceInterval_coverage: observed used

AronowAudit.P047.confidenceInterval_root_width: observed used

AronowAudit.P047.covariateSigma_le: observed used

AronowAudit.P047.estimator_hoeffding: observed used

AronowAudit.P047.estimator_unbiased: observed used

AronowAudit.P047.halfWidth_exponential: observed used

AronowAudit.P047.integrable_armIPW: observed used

AronowAudit.P047.integrable_arm_numerator: observed used

AronowAudit.P047.integrable_potential: observed used

AronowAudit.P047.measurable_arm: observed used

AronowAudit.P047.measurable_armIPW: observed used

AronowAudit.P047.measurable_armPropensity_conditioning: observed used

AronowAudit.P047.measurable_covariate: observed used

AronowAudit.P047.measurable_covariate_conditioning: observed used

AronowAudit.P047.measurable_estimator: observed used

AronowAudit.P047.measurable_potential: observed used

AronowAudit.P047.measurable_score: observed used

AronowAudit.P047.sample_coordinate_hasLaw: observed used

AronowAudit.P047.sample_score_identDistrib: observed used

AronowAudit.P047.sample_scores_independent: observed used

AronowAudit.P047.score_abs_le: observed used

AronowAudit.P047.score_eq_armIPW: observed used

AronowAudit.P047.score_unbiased: observed used

AronowAudit.P047.subgaussian_abs_tail: observed used

AronowAudit.P047.symmetric_interval_coverage: observed used

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

P047-T2

Section 1 (Recapitulation), first bullet, first sentence: uniformly root-n consistent and asymptotically normal estimators when the propensity score is known (no numbered result)

p. 85 (PDF p. 2)

Section 1, first bullet, first sentence, p. 85 (PDF p. 2). The known-propensity ATE estimator is uniformly root-n tight with the explicit exponential bound and has a fixed-law centered normal limit with variance at most B squared over epsilon squared, including zero variance.

Historical component credits

Proved AronowAudit.P047.known_propensity_asymptotics Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This exports the complete T2 conjunction: uniform root-n tightness, its all-n exponential bound, the score variance bound, and pointwise Gaussian convergence, including variance zero. Common B and epsilon are expressly in the target and S4, so they are not an added hypothesis. The verdict is unchanged from claim-stage.json; the broader classification has no adverse fidelity consequence.
Proved AronowAudit.P047.score_memLp Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected asymptotic aggregate theorem already supplies this component or its moment justification.
Proved AronowAudit.P047.score_variance_le Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected asymptotic aggregate theorem already supplies this component or its moment justification.
Proved AronowAudit.P047.estimator_central_limit Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected asymptotic aggregate theorem already supplies this component or its moment justification.
Proved AronowAudit.P047.estimator_root_tail Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected asymptotic aggregate theorem already supplies this component or its moment justification.
Proved AronowAudit.P047.estimator_uniform_root_consistency Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected asymptotic aggregate theorem already supplies this component or its moment justification.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:1ece52f58db87ce6d460d303785aedb930858664e1beb0c834864287072d2f14

Effective claim (source TeX): [Printed: Section 1, p. 85 (PDF p. 2)] 'When a practitioner aims to estimate averages of bounded potential outcomes (e.g., the average treatment effect on a binary outcome) in a setting where both ignorability and positivity are known to hold after adjusting for at least one continuous covariate, the following statements are true: • If the propensity score is known, such as in a randomized controlled trial (RCT), there exist simple estimators that are uniformly root-$n$ consistent and asymptotically normal. Confidence intervals based on these estimators are finite-sample valid and their widths shrink at a root-$n$ rate.' The target is the first sentence of the bullet; the second sentence is P047-T1. [Formal reading. The rejoinder prints no formula and does not define 'uniformly root-$n$ consistent' or 'asymptotically normal'. This is the validators' reading (validator artifact, sections 5 and 8; second validator, sections 3 and 5).] Fix constants $0 < B < \infty$ and $0 < \varepsilon \le 1/2$. Let $\mathcal{P}_{B,\varepsilon}$ be the class of all laws $P$ of one unit $(W, A, Y(0), Y(1))$ with $A \in \{0, 1\}$, ignorability $A \perp Y(a) \mid W$ for $a = 0, 1$, $|Y(a)| \le B$ for $a = 0, 1$ and $\varepsilon \le e_P(W) \le 1 - \varepsilon$ almost surely, where $e_P(w) = P(A = 1 \mid W = w)$ is known. For an i.i.d. sample of size $n$ from $P$, with observed outcomes $Y_i = A_iY_i(1) + (1 - A_i)Y_i(0)$, let $Z_i = \frac{A_iY_i}{e_P(W_i)} - \frac{(1 - A_i)Y_i}{1 - e_P(W_i)}$, $T_n = \frac{1}{n}\sum_{i=1}^{n} Z_i$ and $\tau_P = E_P[Y(1) - Y(0)]$. Then: (a) [uniformly root-$n$ consistent] $\lim_{K \to \infty}\ \sup_{n \ge 1}\ \sup_{P \in \mathcal{P}_{B,\varepsilon}} P\left(\sqrt{n}\,|T_n - \tau_P| > K\right) = 0$; indeed $P\left(\sqrt{n}\,|T_n - \tau_P| \ge K\right) \le 2\exp\left(-\frac{K^2\varepsilon^2}{2B^2}\right)$ for every $n \ge 1$, every $K \ge 0$ and every $P \in \mathcal{P}_{B,\varepsilon}$; (b) [asymptotically normal] for each fixed $P \in \mathcal{P}_{B,\varepsilon}$, $\sqrt{n}\,(T_n - \tau_P)$ converges in distribution, as $n \to \infty$, to $N(0, \sigma_P^2)$ with $\sigma_P^2 = \mathrm{Var}_P(Z) \le B^2/\varepsilon^2$ (the point mass at zero if $\sigma_P^2 = 0$). In (a) each law is paired with the estimator computed from its own known propensity score; in particular (a) holds over any subclass of $\mathcal{P}_{B,\varepsilon}$ that shares one known $e$.

: T2Statement

Theorem: SAP.P047.T2

Statement hash: c4f8ecb738b7bc98ac5ece100ee0012be5f5036d37c5563bc07d9200f25ca94c

Statement definition: SAP.P047.T2Statement

Certificate. 20261104T173313Z

Declaration review: faithful; 2026-11-04T17:56:00Z; 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.P047.estimator_central_limit: observed used

AronowAudit.P047.estimator_root_tail: observed used

AronowAudit.P047.estimator_uniform_root_consistency: observed used

AronowAudit.P047.score_memLp: observed used

AronowAudit.P047.score_variance_le: observed used

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

P047-T3

Section 1 (Recapitulation), second bullet: no uniformly consistent estimator and no honest confidence interval of shrinking width when the propensity score is not known (no numbered result)

pp. 85-86 (PDF pp. 2-3)

Section 1, second bullet, pp. 85–86 (PDF pp. 2–3). The usual measure-theoretic conditions are assumed. With unknown propensity, uniform covariate, binary potential outcomes, joint ignorability and common overlap [1/4,3/4], every sample-only estimator has worst-case error at least 1/2 at tolerance 1/12 for every positive sample size, and uniform consistency is impossible. The concrete baseline S belongs to this model, has propensity 1/2 and effect 1/2. For every asymptotically honest interval at level 1-alpha, 0 < alpha < 1/2, its probability under S of width greater than 1/6 has limit inferior at least 1-2*alpha > 0. Width therefore fails to shrink in probability under S and uniformly over the model. All conclusions hold for every containing class of probability laws in which the mean effect exists, without assumptions on its additional laws. They allow an arbitrary independent random seed and measurable extended-real interval endpoints. Thus leaving both nuisance functions unrestricted precludes the two desired properties; no sufficiency or separate mathematical notion of testability is asserted.

Historical component credits

Proved AronowAudit.P047.Wave1.unknown_propensity_impossibility_randomized Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This proves the printed positive impossibility claim in its explicit target reading, not a counterexample or refutation. The uniform/binary/joint-ignorability subclass is explicitly the target class and suffices for nonexistence over larger unrestricted models. The target expressly leaves the printed adjective untestable without a formal reading; no sufficiency assertion is credited. The record confirms the statement reading and does not change the saved claim-stage verdict.
Proved AronowAudit.P047.cell_overlap Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.cell_mean_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.single_cell_observable_agreement Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.measurable_highIndicator Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.highIndicator_memLp Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.highIndicator_mean Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.highIndicator_variance Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.measurable_highProportion Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.highProportion_moments Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.familyEffect_eq_cell_average Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.collision_union_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.two_point_estimator_algebra Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.separated_bands_force_width Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.highProportion_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.measurable_familyEffect Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.familyEffect_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.familyEffect_memLp Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.familyEffect_moments Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.familyEffect_prior_concentration Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.two_point_interval_algebra Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.uniform_labels_pair_collision Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.iid_uniform_labels_collision Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.event_difference_of_agreement_on_good Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.measurable_baseline_map Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.baseline_covariate_law Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.
Proved AronowAudit.P047.baseline_ate Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The completed randomized impossibility theorem supplies the retained conclusions; this construction helper is not separately invoked.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:17075260aa7c99b1a467cc79bd848fd27e38ac53a96f5122afba52d54407aa89

Effective claim (source TeX): [Printed: Section 1, pp. 85-86 (PDF pp. 2-3); the bullet begins at the foot of p. 85 and ends at the top of p. 86] 'When a practitioner aims to estimate averages of bounded potential outcomes (e.g., the average treatment effect on a binary outcome) in a setting where both ignorability and positivity are known to hold after adjusting for at least one continuous covariate, the following statements are true: [...] • If the propensity score is not known, such as in an observational study, there exist neither uniformly consistent estimators nor uniform (i.e., honest) large-sample confidence intervals whose widths are shrinking with the sample size. To achieve these properties, the practitioner must impose untestable assumptions on either the propensity score function or the conditional expectation function of the outcomes.' [Attribution, p. 86 (PDF p. 3)] 'These insights are decades old, going back to Robins and Ritov (1997), but they are not yet common knowledge.' [Formal reading. The rejoinder prints no formula. This is the form in which the audit's two fresh readers checked the bullet.] Let $\mathcal{P}$ be the class of all laws $P$ of one unit $(W, A, Y(0), Y(1))$ with $W \sim \mathrm{Uniform}(0, 1)$, $A \in \{0, 1\}$, $Y(0), Y(1) \in \{0, 1\}$, ignorability $A \perp (Y(0), Y(1)) \mid W$, and positivity $1/4 \le e_P(W) \le 3/4$ almost surely, where $e_P(w) = P(A = 1 \mid W = w)$. The propensity score $e_P$ and the conditional expectation functions $w \mapsto E_P[Y(a) \mid W = w]$ are otherwise unrestricted measurable functions. The observed outcome is $Y = AY(1) + (1 - A)Y(0)$ and $\tau(P) = E_P[Y(1) - Y(0)]$. The propensity score is not known: an estimator is a sequence of measurable functions $\hat\tau_n$ of the observed sample $(W_i, A_i, Y_i)_{i=1}^{n}$, the same for every $P \in \mathcal{P}$, and an interval procedure is a sequence of intervals $C_n$ whose endpoints are such functions. Then: (a) [no uniformly consistent estimator] for every estimator there is $\epsilon > 0$ with $\limsup_{n \to \infty}\sup_{P \in \mathcal{P}} P\left(|\hat\tau_n - \tau(P)| > \epsilon\right) > 0$; in fact, for every $n \ge 1$ and every estimator, $\sup_{P \in \mathcal{P}} P\left(|\hat\tau_n - \tau(P)| \ge \frac{1}{12}\right) \ge \frac12$; (b) [no honest interval of shrinking width] let $0 < \alpha < 1/2$ and let $C_n$ be an interval procedure with $\liminf_{n \to \infty}\inf_{P \in \mathcal{P}} P\left(\tau(P) \in C_n\right) \ge 1 - \alpha$. Then the width of $C_n$ does not converge to zero in probability under the single law $S \in \mathcal{P}$ that has $e_S \equiv 1/2$, $Y(0) = 0$ and $Y(1) \mid W \sim \mathrm{Bernoulli}(1/2)$ independent of $A$ given $W$: $\liminf_{n \to \infty} S\left(\mathrm{width}(C_n) > \frac16\right) \ge 1 - 2\alpha > 0$. A fortiori the width does not converge to zero in probability uniformly over $\mathcal{P}$; (c) [second sentence of the bullet] its formal content is (a) and (b): with both the propensity score function and the conditional expectation function of the outcomes unrestricted, neither property can be achieved, so a restriction on at least one of the two is necessary. The word 'untestable' is given no formal reading, and no claim that a restriction on one of the two functions is sufficient is part of the target. Since a supremum over a class is at least the supremum over a subclass, (a) and (b) hold for every model that contains $\mathcal{P}$, for example with an arbitrary continuous covariate law, bounded rather than binary outcomes, or a smaller positivity constant.

: SAP.P047.T3WholeStatement.{u}

Theorem: SAP.P047.T3

Statement hash: 359c75fe26f86e5b54d6e4b900c33e214b50f5c187c9ed8680d388d70e290e03

Statement definition: SAP.P047.T3WholeStatement

Certificate. 20261104T173313Z

Declaration review: faithful; 2026-11-04T17:56:00Z; 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.P047.baseline_ate: observed used

AronowAudit.P047.cell_mean_bounds: observed used

AronowAudit.P047.cell_overlap: observed used

AronowAudit.P047.collision_union_bound: observed used

AronowAudit.P047.familyEffect_bounds: observed used

AronowAudit.P047.familyEffect_eq_cell_average: observed used

AronowAudit.P047.familyEffect_memLp: observed used

AronowAudit.P047.familyEffect_moments: observed used

AronowAudit.P047.familyEffect_prior_concentration: observed used

AronowAudit.P047.highIndicator_mean: observed used

AronowAudit.P047.highIndicator_memLp: observed used

AronowAudit.P047.highIndicator_variance: observed used

AronowAudit.P047.highProportion_bounds: observed used

AronowAudit.P047.highProportion_moments: observed used

AronowAudit.P047.measurable_baseline_map: observed used

AronowAudit.P047.measurable_familyEffect: observed used

AronowAudit.P047.measurable_highIndicator: observed used

AronowAudit.P047.measurable_highProportion: observed used

AronowAudit.P047.single_cell_observable_agreement: observed used

AronowAudit.P047.uniform_labels_pair_collision: observed used

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

Reproduction

What I recalculated or reran from public materials. "Agrees", "Agrees in part" or "Disagrees" compares my result with the printed one. "Not compared" means the comparison could not be made.

Status: Not applicable
Agreement: Not applicable