Appendix A.2, Lemma A.1, pp. 1510–1511 (PDF pp. 9–10). The six-unit counterexample has all the stated means, coefficients, moments, expectation, variance, bound and positive gap. Covariance and its variance are unchanged by constant shifts; the printed bound changes under centering, and its centered value bounds the variance.
", "headline": "Lemma A.1, the variance bound, as printed for arbitrary n-vectors a, b. The witness is the validator's and is not in the paper", "location": "p. 1511 (PDF p. 10), Appendix A.2; the vectors a, b and the covariance estimators are introduced on p. 1510 (PDF p. 9)" }, "id": "P005-T1", "text": { "body": "Appendix A.2, Lemma A.1, pp. 1510–1511 (PDF pp. 9–10). The six-unit counterexample has all the stated means, coefficients, moments, expectation, variance, bound and positive gap. Covariance and its variance are unchanged by constant shifts; the printed bound changes under centering, and its centered value bounds the variance.", "headline": "Lemma A.1, the variance bound, as printed for arbitrary $n$-vectors $a$, $b$. The witness is the validator's and is not in the paper", "location": "p. 1511 (PDF p. 10), Appendix A.2; the vectors $a$, $b$ and the covariance estimators are introduced on p. 1510 (PDF p. 9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T1" }, { "declarations": [ { "claim_revision_id": "catalog:5c6443506628f488b4ba342d66180dd1a8ca7057aedd752d50358c6aebfb1ef7", "conditional_on": [ "L. Lei and P. Ding, \"Regression adjustment in completely randomized experiments with a diverging number of covariates\", Biometrika 108 (2021), Supplementary Material, Lemma A.5" ], "credit_reason": "Its literature premise is not a canonical hypothesis; the unconditional centered bound and expectation are instead supplied by existing supporting declarations.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.corrected_covariance_lemma_conditional", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Cond/CondP002P005P006Proofs.lean#L232", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c6443506628f488b4ba342d66180dd1a8ca7057aedd752d50358c6aebfb1ef7", "conditional_on": [], "credit_reason": "This harmless generalization includes exactly the centering operation needed by the correction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.covariance_location_invariant", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L357", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c6443506628f488b4ba342d66180dd1a8ca7057aedd752d50358c6aebfb1ef7", "conditional_on": [], "credit_reason": "Empty-design totalization adds cases but does not replace the nonempty sampling case the target needs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.variance_difference_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L960", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5c6443506628f488b4ba342d66180dd1a8ca7057aedd752d50358c6aebfb1ef7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T2" }, "html": { "body": "Appendix A.2, Lemma A.1, p. 1511 (PDF p. 10). For either nonempty arm, the printed expectation and the variance bound with both vectors centered hold. Covariance is location invariant, the centered sample-product mean has the simple-random-sampling variance, the product of centered sums has the stated exact expectation and variance and their bound, and the generic variance-difference inequality gives the repaired bound. The printed bound is recovered when both population means are zero.
", "headline": "Lemma A.1, as repaired: the variance bound with centred vectors (Sol's smallest repair and the validator's derivation)", "location": "p. 1511 (PDF p. 10), Appendix A.2" }, "id": "P005-T2", "text": { "body": "Appendix A.2, Lemma A.1, p. 1511 (PDF p. 10). For either nonempty arm, the printed expectation and the variance bound with both vectors centered hold. Covariance is location invariant, the centered sample-product mean has the simple-random-sampling variance, the product of centered sums has the stated exact expectation and variance and their bound, and the generic variance-difference inequality gives the repaired bound. The printed bound is recovered when both population means are zero.", "headline": "Lemma A.1, as repaired: the variance bound with centred vectors (Sol's smallest repair and the validator's derivation)", "location": "p. 1511 (PDF p. 10), Appendix A.2" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T2" }, { "declarations": [ { "claim_revision_id": "catalog:8d0005d2108e1ecdd2266b7ee73d1ae0a1ea3a54c00c449dc48a37758167bf2f", "conditional_on": [], "credit_reason": "The credited component is the finite failure of the first two bounds. To satisfy sequence language, insert this n=16 population into the standardized scalar-grid sequence of R037-15; K=1 and fixed allocation bounds remain valid, and all-assignment regularity holds on the relevant tail.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.leverage_variance_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B4b.lean#L264", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:8d0005d2108e1ecdd2266b7ee73d1ae0a1ea3a54c00c449dc48a37758167bf2f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T3", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T3" }, "html": { "body": "Appendix B.2, item 2, p. 1517 (PDF p. 16). The standardized regular sixteen-unit counterexample satisfies all stated moment and allocation conditions. Its leverage vector, mean and constant leverage-product, three-category probabilities and covariance values, exact variance, both failed bounds and positive gaps, and the valid third-expression and centered-bound comparisons have the stated numerical values.
", "headline": "Proof of Theorem 3.2 (Appendix B.2), item 2: the chain “First notice, by Lemma A.1 and Assumption 1, …”, as printed. The witness is the validator's and is not in the paper", "location": "p. 1517 (PDF p. 16), Appendix B.2, item 2" }, "id": "P005-T3", "text": { "body": "Appendix B.2, item 2, p. 1517 (PDF p. 16). The standardized regular sixteen-unit counterexample satisfies all stated moment and allocation conditions. Its leverage vector, mean and constant leverage-product, three-category probabilities and covariance values, exact variance, both failed bounds and positive gaps, and the valid third-expression and centered-bound comparisons have the stated numerical values.", "headline": "Proof of Theorem 3.2 (Appendix B.2), item 2: the chain “First notice, by Lemma A.1 and Assumption 1, …”, as printed. The witness is the validator's and is not in the paper", "location": "p. 1517 (PDF p. 16), Appendix B.2, item 2" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T3" }, { "declarations": [ { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/Gap/GP005B4c.lean:500, the conclusion includes mean(u²) = mean(h²) − d² and mean(u²) ≤ nκd, but omits mean(u²) ≤ mean(h²) and mean(h²) ≤ nκd. P005-T4(b) explicitly asserts both inequalities, and this row claims the whole fixed-n component T4(a)–(e). In particular, the raw-second-moment bound is a separate, stronger conclusion than the exported centered bound; its derivability from the hypotheses does not supply the missing conclusion. The centering, finite-sample assumptions and quantifier over both arms otherwise match the corrected claim.', 'suggested_fix': 'Extend the conclusion with popMean (fun i => u i^2) ≤ popMean (fun i => leverage P i^2) and popMean (fun i => leverage P i^2) ≤ (n : ℝ) * kappa P * d, retaining the existing identity and variance chain.', 'second_review': 'P005/Wave1/Gap/GP005B4c.lean:500 exports the centered second-moment identity and terminal bound but omits the two intermediate inequalities expressly required by P005-T4(b), mean(u²) ≤ mean(h²) ≤ nκd. The omission needs a statement repair, but it is minor: the already exported mean(h) = d and 0 ≤ h_i ≤ nκ give mean(h²) ≤ nκ mean(h) = nκd by averaging h_i² ≤ nκh_i, while the exported identity and d² ≥ 0 give the other inequality. The fixed-n assumptions, both-arm quantifier, and centered variance chain otherwise match the credited T4(a)–(e) component.', 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P005.Wave1.corrected_leverage_variance_chain_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B4c.lean#L500", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/Gap/GP005B5.lean:350, the conclusion gives Big-O bounds only for the actual unscaled and scaled covariance variances. P005-T4(f) additionally asserts that the explicit right-hand side of T4(e) is O(nκ² + κd/n), and states two coefficient comparisons using the allocation lower bound; none of these conclusions is present. Calling these assertions proof details in the earlier record does not remove them from the supplied target. The restriction to positive covariate dimension excludes a trivial zero-leverage case, and finite initial sequence indices do not affect the asymptotic assertions.', 'suggested_fix': 'For a fixed uniform outcome-moment bound K, add IsBigO atTop U_t (fun n => (n : ℝ) * kappa (P n)^2 + kappa (P n) * d n / n), where U_t is the final numerical right-hand side of T4(e) with arm size s_n = armSize n (m n) t. Also export the comparisons 2(n−s)/(s(n−1)) ≤ 2/(a(n−1)) and 2(n−s)n/(s³(n−1)) ≤ 2/(a³n(n−1)) when 0 < a < s/n and n ≥ 4; take a = p_min for treatment and a = 1−p_max for control. Retain the existing variance conclusions.', 'second_review': 'P005/Wave1/Gap/GP005B5.lean:350 exports the actual covariance-variance orders, whereas P005-T4(f) also expressly requires the order of the numerical right-hand side U of (e) and the two allocation coefficient comparisons. Those conclusions are absent, but their repair is elementary: take a = lo for treatment or a = 1−hi for control from P005/P005_Formalization.lean:332; s ≥ an and n−s ≤ n give the stated coefficient bounds, and for n ≥ 4 they yield 0 ≤ U ≤ 4K(a⁻¹+a⁻³)(nκ²+κd/n). Thus the missing order is immediate deterministic arithmetic under the existing assumptions, warranting minor severity; the additional positive-dimension restriction also excludes only the trivial zero-leverage case.', 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P005.Wave1.corrected_leverage_variance_order_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B5.lean#L351", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "conditional_on": [], "credit_reason": "Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.mean_add", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L930", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "conditional_on": [], "credit_reason": "Not invoked separately: its relevant content is already incorporated in the selected primary proof; 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no additional conclusion is asserted from this auxiliary.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.leverage_population_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1211", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "conditional_on": [], "credit_reason": "Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P005.centered_leverage_moment_bounds", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1226", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T4" }, "html": { "body": "Appendix B.2, item 2, p. 1517 (PDF p. 16). For either arm the centered leverage and outcome moments give every inequality in the repaired finite variance chain, including the intermediate raw leverage second moment. The two allocation coefficients obey the stated bounds. Under Assumptions 1–3 the numerical right-hand side and the actual variance are O(nκ²+κd/n), and the variance after division by n is O(κ²/n+κd/n³). Dimension zero is allowed, including along a sequence.
", "headline": "Proof of Theorem 3.2 (Appendix B.2), item 2, as repaired: the variance chain with h and y(1) centred (Sol's smallest repair and the validator's derivation)", "location": "p. 1517 (PDF p. 16), Appendix B.2, item 2" }, "id": "P005-T4", "text": { "body": "Appendix B.2, item 2, p. 1517 (PDF p. 16). For either arm the centered leverage and outcome moments give every inequality in the repaired finite variance chain, including the intermediate raw leverage second moment. The two allocation coefficients obey the stated bounds. Under Assumptions 1–3 the numerical right-hand side and the actual variance are O(nκ²+κd/n), and the variance after division by n is O(κ²/n+κd/n³). Dimension zero is allowed, including along a sequence.", "headline": "Proof of Theorem 3.2 (Appendix B.2), item 2, as repaired: the variance chain with $h$ and $y(1)$ centred (Sol's smallest repair and the validator's derivation)", "location": "p. 1517 (PDF p. 16), Appendix B.2, item 2" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T4" }, { "declarations": [ { "claim_revision_id": "catalog:60484f0e60dc13f45cf1fde1df5fdb3e8d7c9d4bbf30f1cae12da10f09ce33f5", "conditional_on": [], "credit_reason": "The finite third-component identity is contradicted under every required fixed-n condition; insertion at n=6 into a standardized distinct-grid sequence with K=28 also supplies the standing sequence interpretation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.interacted_sign_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L623", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:60484f0e60dc13f45cf1fde1df5fdb3e8d7c9d4bbf30f1cae12da10f09ce33f5", "conditional_on": [], "credit_reason": "Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.design_six_two", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L598", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:60484f0e60dc13f45cf1fde1df5fdb3e8d7c9d4bbf30f1cae12da10f09ce33f5", "conditional_on": [], "credit_reason": "Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.expectation_six_two", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L602", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:60484f0e60dc13f45cf1fde1df5fdb3e8d7c9d4bbf30f1cae12da10f09ce33f5", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T5" }, "html": { "body": "Theorem 4.1, last display, p. 1509 (PDF p. 8). The printed claim is false: the specified six-unit population satisfies the finite-sample assumptions, has positive variances under all fifteen assignments, and gives 217/2500 for the third bias component while the printed formula gives -217/2500. All individual means, moments, coefficients and expectations are included; reversing both printed signs gives the actual component.
", "headline": "Theorem 4.1, last display on p. 1509 (the third bias component E[xˉC′ν3C]−E[xˉT′ν3T]), as printed. The witness is the validator's and is not in the paper", "location": "p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page" }, "id": "P005-T5", "text": { "body": "Theorem 4.1, last display, p. 1509 (PDF p. 8). The printed claim is false: the specified six-unit population satisfies the finite-sample assumptions, has positive variances under all fifteen assignments, and gives 217/2500 for the third bias component while the printed formula gives -217/2500. All individual means, moments, coefficients and expectations are included; reversing both printed signs gives the actual component.", "headline": "Theorem 4.1, last display on p. 1509 (the third bias component $E[\\bar x_C'\\nu_{3C}] - E[\\bar x_T'\\nu_{3T}]$), as printed. The witness is the validator's and is not in the paper", "location": "p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T5" }, { "declarations": [ { "claim_revision_id": "catalog:f81f48270dcc67ee96590c34f530409cf9e548487564c90807873dc818e3574a", "conditional_on": [], "credit_reason": "The negative arm identities omitted as separate conjuncts are algebraic restatements of the two exported moments, so no independent conclusion is lost. No corrected-sign assertion is mistaken for the printed identity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.corrected_interacted_third_bias_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B5.lean#L678", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f81f48270dcc67ee96590c34f530409cf9e548487564c90807873dc818e3574a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T6", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T6" }, "html": { "body": "Theorem 4.1, last display, p. 1509 (PDF p. 8). For centered covariates, both third-moment identities, their signed bias components and the corrected control-minus-treatment formula hold exactly.
", "headline": "Theorem 4.1, last display on p. 1509, as repaired: both signs of the right-hand side reversed (Sol's smallest repair)", "location": "p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page" }, "id": "P005-T6", "text": { "body": "Theorem 4.1, last display, p. 1509 (PDF p. 8). For centered covariates, both third-moment identities, their signed bias components and the corrected control-minus-treatment formula hold exactly.", "headline": "Theorem 4.1, last display on p. 1509, as repaired: both signs of the right-hand side reversed (Sol's smallest repair)", "location": "p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T6" }, { "declarations": [ { "claim_revision_id": "catalog:ced1bdd8e67a9027a75e122bb092cb8fcb6b82a527a44c1860c84fb51a0b4983", "conditional_on": [], "credit_reason": "Eventual hypotheses and the even-subsequence lower bound give a genuine failure of full-sequence tightness. The fixed-n exact maxima and illustrative enumerations in the target are diagnostics rather than additional requirements on this witness.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.univariate_endpoint_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GLP005B2.lean#L109", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ced1bdd8e67a9027a75e122bb092cb8fcb6b82a527a44c1860c84fb51a0b4983", "conditional_on": [], "credit_reason": "A simple auxiliary identity can be faithful even though proving it is elementary; the actual asymptotic counterexample is separately exported by R037-13.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.univariate_printed_condition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L922", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ced1bdd8e67a9027a75e122bb092cb8fcb6b82a527a44c1860c84fb51a0b4983", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T7", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T7" }, "html": { "body": "Theorem 4.1, pp. 1509--1510 (PDF pp. 8--9), footnote 8. The printed claim is false: the explicit one-covariate spike sequence satisfies Assumptions 1--3 and every-assignment regularity, yet its remainder is not of the printed stochastic order. The statement includes all grid identities, pointwise bounds, exact leverage and slopes, fitted-line calculations, sharp bounds, probability-one even-size tails and both exact finite minima.
", "headline": "Theorem 4.1, final paragraph: the stochastic expansion under “κlogd=o(1)”, as printed, at d=1. The sequence is the validator's and is not in the paper", "location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1 (the theorem begins on p. 1509, PDF p. 8), with footnote 8" }, "id": "P005-T7", "text": { "body": "Theorem 4.1, pp. 1509--1510 (PDF pp. 8--9), footnote 8. The printed claim is false: the explicit one-covariate spike sequence satisfies Assumptions 1--3 and every-assignment regularity, yet its remainder is not of the printed stochastic order. The statement includes all grid identities, pointwise bounds, exact leverage and slopes, fitted-line calculations, sharp bounds, probability-one even-size tails and both exact finite minima.", "headline": "Theorem 4.1, final paragraph: the stochastic expansion under “$\\kappa\\log d = o(1)$”, as printed, at $d = 1$. The sequence is the validator's and is not in the paper", "location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1 (the theorem begins on p. 1509, PDF p. 8), with footnote 8" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T7" }, { "declarations": [ { "claim_revision_id": "catalog:73cfad2844215a48f4019f8cb6c0724d153ec400be483e98756af197c7885aae", "conditional_on": [ "J. A. Tropp, \"Improved analysis of the subsampled randomized Hadamard transform\", Advances in Adaptive Data Analysis 3 (2011), Lemma 3.4, pp. 115-126" ], "credit_reason": "The only credited expansion theorem requires the unproved proposition Tropp2011RowSampling. The canonical claim has no such hypothesis, and no unconditional estimator expansion is available from this row. Do not turn the cited proposition into an assumption. The separately targeted logarithm equivalences are retained in T9.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.corrected_interacted_expansion_max_conditional", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Cond/CondP005T8T9Proofs.lean#L709", "status": "proved", "support_scope": "historical declaration", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:73cfad2844215a48f4019f8cb6c0724d153ec400be483e98756af197c7885aae", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T8", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T8" }, "html": { "body": "No current theorem is supplied for this target.
", "headline": "Theorem 4.1, final paragraph, as repaired: the condition κlogmax{d,2}=o(1) (Sol's smallest repair forP005-E001)",
"location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)"
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"body": "No current theorem is supplied for this target.",
"headline": "Theorem 4.1, final paragraph, as repaired: the condition $\\kappa\\log\\max\\{d,2\\} = o(1)$ (Sol's smallest repair for P005-E001)",
"location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)"
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"body": "Theorem 4.1, p. 1510 (PDF p. 9), and Appendix B.3, p. 1517 (PDF p. 16). Under Assumptions 1--3 and every-assignment invertibility, the repaired condition kappa log(2d) tending to zero implies the full interacted expansion with remainder O_p(sqrt(kappa d/n)). The two repaired logarithms have the stated inequalities and equivalent limits. The repaired condition implies vanishing leverage and the printed condition, and is equivalent to the printed condition plus vanishing leverage on the dimension-one subsequence.
", "headline": "Theorem 4.1, final paragraph, as repaired in the form given by the validator ofP005-N004: the condition κlog(2d)=o(1), with its equivalence to the form of P005-T8",
"location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)"
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"body": "Theorem 4.1, p. 1510 (PDF p. 9), and Appendix B.3, p. 1517 (PDF p. 16). Under Assumptions 1--3 and every-assignment invertibility, the repaired condition kappa log(2d) tending to zero implies the full interacted expansion with remainder O_p(sqrt(kappa d/n)). The two repaired logarithms have the stated inequalities and equivalent limits. The repaired condition implies vanishing leverage and the printed condition, and is equivalent to the printed condition plus vanishing leverage on the dimension-one subsequence.",
"headline": "Theorem 4.1, final paragraph, as repaired in the form given by the validator of P005-N004: the condition $\\kappa\\log(2d) = o(1)$, with its equivalence to the form of P005-T8",
"location": "p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)"
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"body": "Appendix B.4, last display, pp. 1518--1519 (PDF pp. 17--18). The printed claim is false: the six-unit standardized regular population has printed treated expectation 4913/1200 instead of 20/3, while the derived weight is unbiased. All fifteen-assignment positivity, leverage, population covariance, assignmentwise and expected gaps, and rational scaling identities are included.
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"body": "Appendix B.4, corrected last display, pp. 1518--1519 (PDF pp. 17--18). The corrected arm expressions cancel the inverse assignmentwise, have the exact generic expectation, and are unbiased with the derived arm-size weights. The cancelling treated weight equals its unsimplified expression and is unique for arms of size at least two; singleton expectations require no division by s-1. The printed treated weight has the stated bias. Standardization gives d/n <= kappa <= 1 and h_i <= n. Under Assumptions 1--3 both corrected arm remainders are O_p(sqrt(kappa/n)), from any starting size, with no extra rate assumption.",
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"body": "Appendix B.4, display introduced by “Similarly”, p. 1518 (PDF p. 17). The printed claim is false: for the six-unit population with a=(-3,-2,-1,0,1,5), x=sqrt(3/20)a, y(1)=0 and y(0)=a², all 15 assignments have positive treated and control Gram matrices. The population is standardized, has bounded second moments and allocation 1/3, and has h_i=(3/20)a_i² and mean leverage one. The control slope decomposes into its population slope and the three nu terms; the mean control covariate and its population-slope product have expectation zero. The nuisance-product and fitted-slope-product expectations both equal 31085815448171/17062566567774 (approximately 1.822). The printed estimator has expectation 12010469/846720 (approximately 14.18), so is biased; replacing the treated inverse by the control inverse gives exactly the nuisance expectation. The arm sizes, variance scaling, coefficient 1/8 and all rational-coordinate identities hold assignment by assignment.
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"body": "Appendix B.2, pp. 1515–1516 (PDF pp. 14–15), and Remark 2, p. 1507 (PDF p. 6). The corrected debiased expansion, inverse cancellation, arbitrary-vector residual identities, centered arm-mean identities and actual bias decomposition hold. A standardized everywhere regular four-unit example shows that the printed treated-set control sum can differ from the control residual mean. One function of the assignment, covariates and observed outcomes, chosen before the potential outcomes, gives the inverse bias component.
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"html": {
"body": "Appendix B.1.1, top of p. 1514 (PDF p. 13), corrected brackets. The weighted normal equation gives all three bracketed vectors and identifies their first expression with the actual common-residual term (*).
", "headline": "Appendix B.1.1, the three-line equality for the term (∗) at the top of p. 1514, as repaired: the unmatched closing parenthesis of the second line deleted (the part ofP005-N005 that is established)",
"location": "p. 1514 (PDF p. 13), Appendix B.1.1, first display and the sentence after it"
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"id": "P005-T18",
"text": {
"body": "Appendix B.1.1, top of p. 1514 (PDF p. 13), corrected brackets. The weighted normal equation gives all three bracketed vectors and identifies their first expression with the actual common-residual term (*).",
"headline": "Appendix B.1.1, the three-line equality for the term $(*)$ at the top of p. 1514, as repaired: the unmatched closing parenthesis of the second line deleted (the part of P005-N005 that is established)",
"location": "p. 1514 (PDF p. 13), Appendix B.1.1, first display and the sentence after it"
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"credit_reason": "This negates the conjunction by its variance clause; it neither negates the valid expectation clause separately nor proves the corrected centered covariance bound.",
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"html": {
"body": "The printed claim is false: the universal conjunction of the printed expectation identity and the uncentered covariance variance bound fails under complete randomization. Lemma A.1, pp. 1510–1511 (PDF pp. 9–10).
", "headline": "Lemma A.1, as printed", "location": "p. 1511 (PDF p. 10), Appendix A.2; definitions on p. 1510 (PDF p. 9)" }, "id": "P005-T19", "text": { "body": "The printed claim is false: the universal conjunction of the printed expectation identity and the uncentered covariance variance bound fails under complete randomization. Lemma A.1, pp. 1510–1511 (PDF pp. 9–10).", "headline": "Lemma A.1, as printed", "location": "p. 1511 (PDF p. 10), Appendix A.2; definitions on p. 1510 (PDF p. 9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T19" }, { "declarations": [ { "claim_revision_id": "catalog:5e4541e5319b1b2a14e738482a3aa167ba6eef808b369b96e424045926d2bd5d", "conditional_on": [], "credit_reason": "Only the valid exact finite-sample portion of T20 is certified here. The n>=2 formulation and the admitted d=0 case exclude no required instance; the nonvacuity example has positive dimension.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.interacted_exact_decomposition_completion", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/ExactDecomposition.lean#L21", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5e4541e5319b1b2a14e738482a3aa167ba6eef808b369b96e424045926d2bd5d", "conditional_on": [], "credit_reason": "The finite sign-identity component is refuted with the printed negative definition of armNu3. Other valid components of T20 receive no refutation credit from this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.ConfidenceFollowup02.printed_interacted_third_bias_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_ConfidenceCompletion.lean#L28", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5e4541e5319b1b2a14e738482a3aa167ba6eef808b369b96e424045926d2bd5d", "conditional_on": [], "credit_reason": "Metadata inconsistency: claims.json marks T20 asymptotic=false although its statement includes the expansion and this row is explicitly credited to refute it. The record target note calls T20 finite-sample-only and assigns the expansion to T7/T8, conflicting with the packet statement and row attribution. This scope issue does not change the fidelity of the displayed negation. No claim-stage verdict changed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.printed_interacted_expansion_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/InteractedExpansion.lean#L22", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5e4541e5319b1b2a14e738482a3aa167ba6eef808b369b96e424045926d2bd5d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T20", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T20" }, "html": { "body": "The printed claim is false: both the universal third-component sign identity and the stochastic expansion under the literal logarithm condition fail; the stated exact slope decompositions, bias expression, and first-component moments hold. Theorem 4.1, pp. 1509–1510 (PDF pp. 8–9).
", "headline": "Theorem 4.1, as printed", "location": "pp. 1509–1510 (PDF pp. 8–9), Section 4; proof B.3 on p. 1517 (PDF p. 16)" }, "id": "P005-T20", "text": { "body": "The printed claim is false: both the universal third-component sign identity and the stochastic expansion under the literal logarithm condition fail; the stated exact slope decompositions, bias expression, and first-component moments hold. Theorem 4.1, pp. 1509–1510 (PDF pp. 8–9).", "headline": "Theorem 4.1, as printed", "location": "pp. 1509–1510 (PDF pp. 8–9), Section 4; proof B.3 on p. 1517 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T20" }, { "declarations": [ { "claim_revision_id": "catalog:afd7a73facb0408e4462811202449af1aceb00bfdc3bbaac5fc696cfd5ee11da", "conditional_on": [], "credit_reason": "The target explicitly incorporates n>=3, nonempty arms, fixed-n Assumption 2 and footnote-4 all-assignment invertibility; none is an added restriction. The separate generalized-inverse extension in the footnote is outside the nonsingular component credited to this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.noninteracted_exact_unbiasedness_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B3b.lean#L151", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:afd7a73facb0408e4462811202449af1aceb00bfdc3bbaac5fc696cfd5ee11da", "conditional_on": [], "credit_reason": "The exact-unbiasedness row already incorporates the sample-mean identity; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.mean_unbiased", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1171", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:afd7a73facb0408e4462811202449af1aceb00bfdc3bbaac5fc696cfd5ee11da", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T21", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T21" }, "html": { "body": "Theorem 3.2, pp. 1508–1509 (PDF pp. 7–8); proof in Appendix B.2. The printed noninteracted bias estimator is exactly unbiased for the bias, and its correction is exactly unbiased for the ATE. The closed-form coefficients are the unique OLS minimizer. The inverse-free formula, bias decomposition, inverse-component identity and both population component expectations hold for all nonempty arms. Expressions (11) and (12), their expectations and their collection into (7) require only their presupposed nonzero arm-size denominators.
", "headline": "Theorem 3.2, first part: exact unbiasedness of Bias and of ATEDebiased (noninteracted case)", "location": "pp. 1508–1509 (PDF pp. 7–8), Section 3, equations (7)–(8); proof in Appendix B.2, pp. 1515–1516 (PDF pp. 14–15)" }, "id": "P005-T21", "text": { "body": "Theorem 3.2, pp. 1508–1509 (PDF pp. 7–8); proof in Appendix B.2. The printed noninteracted bias estimator is exactly unbiased for the bias, and its correction is exactly unbiased for the ATE. The closed-form coefficients are the unique OLS minimizer. The inverse-free formula, bias decomposition, inverse-component identity and both population component expectations hold for all nonempty arms. Expressions (11) and (12), their expectations and their collection into (7) require only their presupposed nonzero arm-size denominators.", "headline": "Theorem 3.2, first part: exact unbiasedness of $\\widehat{\\mathrm{Bias}}$ and of $\\widehat{\\mathrm{ATE}}_{\\mathrm{Debiased}}$ (noninteracted case)", "location": "pp. 1508–1509 (PDF pp. 7–8), Section 3, equations (7)–(8); proof in Appendix B.2, pp. 1515–1516 (PDF pp. 14–15)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T21" }, { "declarations": [ { "claim_revision_id": "catalog:520f9aa2afd09ae071a9fe0a81af2f0109bc34e7344d3cbba114be09c5b625f6", "conditional_on": [], "credit_reason": "Both armwise conclusions are exported, in addition to the two estimator-unbiasedness conclusions, under the target common normalized nonsingular regime. No asymptotic sequel is credited here.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.interacted_exact_unbiasedness_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B3b.lean#L179", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:520f9aa2afd09ae071a9fe0a81af2f0109bc34e7344d3cbba114be09c5b625f6", "conditional_on": [], "credit_reason": "This is the cancellation component alone, not a positive unbiasedness theorem. The extended algebraic domain loses no target cases.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.interacted_debiased_cancellation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L411", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:520f9aa2afd09ae071a9fe0a81af2f0109bc34e7344d3cbba114be09c5b625f6", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T22", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T22" }, "html": { "body": "Theorem 4.2, p. 1510 (PDF p. 9). The interacted bias correction and corrected ATE are exactly unbiased, with the inverse-free identity and both armwise expectations, including the stated weaker centered-covariate hypotheses.
", "headline": "Theorem 4.2, first part: exact unbiasedness of BiasI and of ATEI,Debiased (interacted case)", "location": "p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, p. 1518 (PDF p. 17)" }, "id": "P005-T22", "text": { "body": "Theorem 4.2, p. 1510 (PDF p. 9). The interacted bias correction and corrected ATE are exactly unbiased, with the inverse-free identity and both armwise expectations, including the stated weaker centered-covariate hypotheses.", "headline": "Theorem 4.2, first part: exact unbiasedness of $\\widehat{\\mathrm{Bias}}_I$ and of $\\widehat{\\mathrm{ATE}}_{I,\\mathrm{Debiased}}$ (interacted case)", "location": "p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, p. 1518 (PDF p. 17)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T22" }, { "declarations": [ { "claim_revision_id": "catalog:ad200e42f434f249ba1d79ca9f62f190a05913e854c781e77a5845bd9e9f5dee", "conditional_on": [], "credit_reason": "The exact fixed-n target and both vanishing cases are exported; the separate stochastic expansion is covered by R040-20 rather than asserted here.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.noninteracted_exact_decomposition_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B2.lean#L790", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ad200e42f434f249ba1d79ca9f62f190a05913e854c781e77a5845bd9e9f5dee", "conditional_on": [], "credit_reason": "The selected decomposition row already contains the slope decomposition.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.noninteracted_slope_decomposition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1550", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ad200e42f434f249ba1d79ca9f62f190a05913e854c781e77a5845bd9e9f5dee", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T23" }, "html": { "body": "Theorem 3.1 and Remark 1, p. 1507 (PDF p. 6). The exact noninteracted slope and bias decompositions give (5), (6), the covariance interpretation, and the two stated vanishing consequences of Remark 1.
", "headline": "Theorem 3.1, exact part: the decomposition of β and the bias components (5) and (6); with the two exact consequences stated in Remark 1", "location": "p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1, pp. 1512–1513 (PDF pp. 11–12), and B.2, p. 1515 (PDF p. 14)" }, "id": "P005-T23", "text": { "body": "Theorem 3.1 and Remark 1, p. 1507 (PDF p. 6). The exact noninteracted slope and bias decompositions give (5), (6), the covariance interpretation, and the two stated vanishing consequences of Remark 1.", "headline": "Theorem 3.1, exact part: the decomposition of $\\widehat\\beta$ and the bias components (5) and (6); with the two exact consequences stated in Remark 1", "location": "p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1, pp. 1512–1513 (PDF pp. 11–12), and B.2, p. 1515 (PDF p. 14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T23" }, { "declarations": [ { "claim_revision_id": "catalog:43328987c47c5d6f08d8faeb6c298bbc8436cede57b4e1bc5e7e8590d80d90a9", "conditional_on": [], "credit_reason": "Covers all of T24, which expressly excludes the separately targeted third-component formula and stochastic expansion. The finite target itself supplies normalization and all-assignment invertibility, so these are not added restrictions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.interacted_exact_decomposition_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B3b.lean#L201", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:43328987c47c5d6f08d8faeb6c298bbc8436cede57b4e1bc5e7e8590d80d90a9", "conditional_on": [], "credit_reason": "The selected decomposition row already contains the two arm slope and numerator identities.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.armN_decomposition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1503", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:43328987c47c5d6f08d8faeb6c298bbc8436cede57b4e1bc5e7e8590d80d90a9", "conditional_on": [], "credit_reason": "The selected decomposition row already contains the two arm slope and numerator identities.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.arm_slope_decomposition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/P005_Formalization.lean#L1535", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:43328987c47c5d6f08d8faeb6c298bbc8436cede57b4e1bc5e7e8590d80d90a9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T24", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T24" }, "html": { "body": "Theorem 4.1, p. 1509 (PDF p. 8), its valid exact part. Both interacted slopes decompose as printed, and the bias expression and separate first-component expectations give their stated difference.
", "headline": "Theorem 4.1, exact part that stands as printed: the decompositions of βI,T and βI,C, the expression of the bias, and the first bias component", "location": "p. 1509 (PDF p. 8), Section 4; proof B.3 on p. 1517 (PDF p. 16) omits the details" }, "id": "P005-T24", "text": { "body": "Theorem 4.1, p. 1509 (PDF p. 8), its valid exact part. Both interacted slopes decompose as printed, and the bias expression and separate first-component expectations give their stated difference.", "headline": "Theorem 4.1, exact part that stands as printed: the decompositions of $\\widehat\\beta_{I,T}$ and $\\widehat\\beta_{I,C}$, the expression of the bias, and the first bias component", "location": "p. 1509 (PDF p. 8), Section 4; proof B.3 on p. 1517 (PDF p. 16) omits the details" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T24" }, { "declarations": [ { "claim_revision_id": "catalog:7e44f8b6f0c09ea443e0e50e7a6584574378c215b25dff1f9aa00640a7c5ee19", "conditional_on": [], "credit_reason": "The target definitions explicitly require 1<=d_n, so the positive-dimension field is not an added assumption. Uniform K, eventual standardization and eventual regularity have the required all-sequence meaning; finitely many initial sizes do not affect O_p.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P005.Wave1.noninteracted_stochastic_expansion_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/Gap/GP005B3b.lean#L847", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:7e44f8b6f0c09ea443e0e50e7a6584574378c215b25dff1f9aa00640a7c5ee19", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T25", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T25" }, "html": { "body": "last part of Theorem 3.1, p. 1507 (PDF p. 6). The noninteracted regression remainder is O_p(sqrt(kappa*d/n)) when d/n tends to zero. The dimension convention 1 <= d_n <= n agrees with the canonical order-symbol definitions. No conclusion is omitted.
", "headline": "Theorem 3.1, last part: the stochastic expansion of ATE (noninteracted case)", "location": "p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1.1, pp. 1513–1515 (PDF pp. 12–14)" }, "id": "P005-T25", "text": { "body": "last part of Theorem 3.1, p. 1507 (PDF p. 6). The noninteracted regression remainder is O_p(sqrt(kappa*d/n)) when d/n tends to zero. The dimension convention 1 <= d_n <= n agrees with the canonical order-symbol definitions. No conclusion is omitted.", "headline": "Theorem 3.1, last part: the stochastic expansion of $\\widehat{\\mathrm{ATE}}$ (noninteracted case)", "location": "p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1.1, pp. 1513–1515 (PDF pp. 12–14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T25" }, { "declarations": [ { "claim_revision_id": "catalog:67ec9f0dfa9acc19273241cedede466b72a9ba4038ce231fcbe9fce372866b82", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/DebiasedLinearTailC.lean:130, the declaration assumes hκ (κ_n → 0) and concludes only that R_n = debiasedATE − ate − commonLeading is o_p(n^−1/2). The final sentence of the supplied P005-T26 claim additionally asserts R_n = O_p(√(κ_n/n)) without that assumption, and identifies R_n exactly with the printed term (*) on p. 1516. Neither the stronger rate without hκ nor the exact identity is included in this declaration’s conclusion. The printed Theorem 3.2 little-o formula is matched, but the supplied target and the row’s full-coverage description assert more.', 'suggested_fix': 'Extend the statement to export stochasticO m R (fun n => Real.sqrt (kappa (P n) / n)) under standingAssumptions and eventual noninteractedRegular without hκ, with the existing stochasticLittleO conclusion conditional on hκ. Also export the fixed-n identity R = −dot (xMean P S − xMean P (control S)) (Nhat P m S − N P m) + (sampleCov S (leverage P) P.yT − sampleCov (control S) (leverage P) P.yC) / (n−2) for n ≥ 3, standardized P, valid arm sizes, S in design n m and nonzero determinant of Lhat.', 'second_review': \"P005/Wave1/DebiasedLinearTailC.lean:130 assumes κ_n → 0 and exports only stochasticLittleO at scale sqrt(1/n), with the residual and randomization interpretation matching the printed expansion through P005/P005_Formalization.lean:310 and P005/P005_Formalization.lean:325. P005-T26's claim text additionally asserts O_p(sqrt(κ_n/n)) without κ_n → 0 and an exact identification with (*); neither conclusion is exported. The little-o conclusion cannot recover that finer rate or cover nonvanishing-leverage sequences, so this is a major statement omission, even though the printed Theorem 3.2 formula itself is represented correctly.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P005.Wave1.noninteracted_debiased_linear_expansion_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/DebiasedLinearTailC.lean#L130", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:67ec9f0dfa9acc19273241cedede466b72a9ba4038ce231fcbe9fce372866b82", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T26", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T26" }, "html": { "body": "Theorem 3.2, p. 1509 (PDF p. 8); Appendix B.2, pp. 1516–1517. The debiased noninteracted remainder equals the p. 1516 term (*) exactly and is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, this gives the printed common-slope expansion with o_p(n^(-1/2)) error.
", "headline": "Theorem 3.2, second part: the asymptotic linear expansion of ATEDebiased", "location": "p. 1509 (PDF p. 8), Section 3 (the theorem begins on p. 1508, PDF p. 7); proof in Appendix B.2, pp. 1516–1517 (PDF pp. 15–16)" }, "id": "P005-T26", "text": { "body": "Theorem 3.2, p. 1509 (PDF p. 8); Appendix B.2, pp. 1516–1517. The debiased noninteracted remainder equals the p. 1516 term (*) exactly and is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, this gives the printed common-slope expansion with o_p(n^(-1/2)) error.", "headline": "Theorem 3.2, second part: the asymptotic linear expansion of $\\widehat{\\mathrm{ATE}}_{\\mathrm{Debiased}}$", "location": "p. 1509 (PDF p. 8), Section 3 (the theorem begins on p. 1508, PDF p. 7); proof in Appendix B.2, pp. 1516–1517 (PDF pp. 15–16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T26" }, { "declarations": [ { "claim_revision_id": "catalog:88011ec1b109ee929518f754f0f34f6f8a2a152edc2046b72f99a6a1944a14cd", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/DebiasedLinearTailC.lean:195, the declaration assumes hκ (κ_n → 0) and concludes only that R_n = interactedDebiasedATE − ate − armLeading is o_p(n^−1/2). The final sentence of the supplied P005-T27 claim also asserts R_n = O_p(√(κ_n/n)) without assuming κ_n → 0. That stronger rate and its wider sequence domain are absent from the declaration, although the printed Theorem 4.2 little-o formula is represented correctly.', 'suggested_fix': 'Extend the statement to export stochasticO m R (fun n => Real.sqrt (kappa (P n) / n)) under standingAssumptions and eventual interactedRegular without hκ, retaining the existing stochasticLittleO conclusion as conditional on hκ.', 'second_review': \"P005/Wave1/DebiasedLinearTailC.lean:195 assumes κ_n → 0 and concludes only stochasticLittleO for interactedDebiasedATE − ate − armLeading at scale sqrt(1/n); armLeading uses the target's two population arm slopes (P005/P005_Formalization.lean:302 and P005/P005_Formalization.lean:313). P005-T27's claim text also requires O_p(sqrt(κ_n/n)) without the decay assumption. That finer rate on the wider sequence domain does not follow from the exported little-o statement, so the target needs a substantive additional statement rather than a description edit.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P005.Wave1.interacted_debiased_linear_expansion_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P005/Wave1/DebiasedLinearTailC.lean#L197", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:88011ec1b109ee929518f754f0f34f6f8a2a152edc2046b72f99a6a1944a14cd", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P005-T27", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P005-T27" }, "html": { "body": "Theorem 4.2, p. 1510 (PDF p. 9); Appendix B.4, pp. 1518–1519. The debiased interacted remainder is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, the printed arm-specific linear expansion has o_p(n^(-1/2)) error.
", "headline": "Theorem 4.2, second part: the asymptotic linear expansion of ATEI,Debiased", "location": "p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, pp. 1518–1519 (PDF pp. 17–18)" }, "id": "P005-T27", "text": { "body": "Theorem 4.2, p. 1510 (PDF p. 9); Appendix B.4, pp. 1518–1519. The debiased interacted remainder is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, the printed arm-specific linear expansion has o_p(n^(-1/2)) error.", "headline": "Theorem 4.2, second part: the asymptotic linear expansion of $\\widehat{\\mathrm{ATE}}_{I,\\mathrm{Debiased}}$", "location": "p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, pp. 1518–1519 (PDF pp. 17–18)" }, "url": "https://pmaronow.github.io/llm-audit/works/P005/#P005-T27" } ] }, "missing_sources": [], "reproduction": { "agreement": "not_applicable", "status": "not_applicable", "targets": [] }, "summary": "The article addresses Freedman's objection that regression adjustment is biased in completely randomized experiments. It derives closed-form estimators of the bias of the noninteracted and interacted OLS estimators of the average treatment effect. Its results are that the bias-corrected estimators are exactly unbiased and, under Freedman's assumptions, have the limiting distributions of the uncorrected ones. 16 entries are classified as errors (1 Moderate, 15 Minor), and 2 are clarity notes. These are counts of entries: \\fid{P005-N004} expands the discussion of the same defect as \\fid{P005-E001} and adds no separate error. The most consequential error is in the last display of Theorem 4.1, which prints the third component of the bias with both signs reversed (\\fid{P005-E003}). The bias correction of Theorem 4.2 has the correct signs.", "what_was_checked": "I checked the following points, which held up. This list is not a statement that the article is correct, and the absence of a correction elsewhere is not a finding of correctness.\n\\begin{itemize}\n\\item The bias estimators of Theorems 3.2 and 4.2 were derived again from the moments of simple random sampling. Enumeration of every assignment confirmed unbiasedness to numerical precision in several small designs and in the twelve designs with $n=24$ and $n_T=8$ of the replication files.\n\\item In Theorem 3.1, the decomposition $\\widehat\\beta=\\beta^*+\\nu_1+\\nu_2+\\nu_3$ and equations (5) and (6) were derived again and agree with enumeration. In Theorem 4.1, the same holds for the decomposition and the first display. The last display is corrected in \\fid{P005-E003}.\n\\item The expectation identity of Lemma A.1 was derived again for vectors that need not be centered. Lemma A.2 and each step of its proof were checked, apart from the misprints corrected in \\fid{P005-E005}, and so were the constants $N_{\\mathrm{TTT}}$, $N_{\\mathrm{CCC}}$ and $N_{\\mathrm{TTC}}$.\n\\item The stochastic expansions of Theorems 3.1, 3.2 and 4.2 were followed step by step through Appendices B.1.1, B.2 and B.4. The stated rates hold once Lemma A.1 is used in its centered form (\\fid{P005-E002}). The expansion of Theorem 4.1 was derived for $d\\ge2$; the case $d=1$ is corrected in \\fid{P005-E001}, and \\fid{P005-N004} concerns the same defect.\n\\item Remarks 1 and 4, the calculation with one covariate on pp.~1507--1508, and footnotes 3 and 4.\n\\item In the replication files, the functions \\texttt{unbiased.reg.noninteract\\_new\\_formula} and \\texttt{unbiased.reg.interact\\_appl\\_new\\_formula} agree with the estimators of Theorems 3.2 and 4.2 on the enumerated designs.\n\\end{itemize}\n\nThe cited works were not among the sources and were not checked. The statements taken from Lei and Ding (2021), Freedman (2008b), Wu and Ding (2021), Finucan, Galbraith and Stone (1974) and Lin (2013) were derived again or checked numerically where the argument needs them, and they were not compared with their sources. The results of Lin (2013) on which the conclusion of the article relies were not examined. The statement in footnote 1 and Remark 5 that earlier corrections are not exactly unbiased was not checked. The simulations of the replication files were not run in full, the designs with $n=36$ were not enumerated, and the scripts in the folder of old files were examined through their differences from the current scripts and targeted checks.\n\nThe article contains no table, figure, simulation or empirical estimate, so there was no numerical result to reproduce. No supplement or correction notice was among the sources. The replication files are a snapshot, dated June 2023, of a public repository of research code. The repository is not a journal replication package and predates publication, and its simulations and applications do not appear in the article. My checks of that code cover the selected formulas and implementation errors reported here.", "work": { "arxiv_id": "2110.08425", "audit_complete": true, "authors": [ "Haoge Chang", "Joel A. Middleton", "P. M. Aronow" ], "category": "Articles - Theory and Methodology", "citation": "Haoge Chang, Joel A. Middleton, and P. M. Aronow (2024). Exact Bias Correction for Linear Adjustment of Randomized Controlled Trials. Econometrica 92(5):1503–1519.", "doi": "10.3982/ECTA20289", "edition": { "pages_in_audited_file": 18, "sha256": "3dab5dd9d1c112bb75028fa4f81cf1e9da3142d48327697e24a4f1c0778e3f38", "source_url": "https://doi.org/10.3982/ECTA20289", "venue": "Econometrica", "year": "2024" }, "id": "P005", "later_materials": null, "title": "Exact Bias Correction for Linear Adjustment of Randomized Controlled Trials", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P005/", "venue": "Econometrica", "year": "2024" } }