Verification record · AUDIT-SCORE-ORACLE-ROBUSTNESS

Independent audit of the frozen-rule scalar retention plug-in CLT (O6)

Verification recordpublication: internalprior art found

Statement

Independent adversarial audit of RETENTION-PLUGIN-CLT-FROZEN-SCALAR and RETENTION-PLUGIN-COVERAGE-DOOR3 at score-oracle-robustness@a5f905e. Verdict: verified with hardened assumptions. O6.1 (RSS identity, 0/0 := 0, library agreement), O6.2 (delta-method CLT with psi = ((1 - eta) S^2 - (S - c_Z)^2)/v and the numerator-denominator covariance form) and O6.3 (a.s. consistency of the plug-in influence variance through within-cell moments of order <= 4) hold as stated; the empty-cell transfer is complete and is replaced by applying the delta method to the estimator's own everywhere-defined functional. Hardening (H1): the sigma^2 = 0 characterisation is correct as a root-set statement but its 'at most two atoms per cell' reading and the sentence 'an atomless cell of positive probability implies (A4)' fail at eta = 0, where the root equation is the identity and every law with vanishing cell means has psi = 0 (CE-O6-ETA-ZERO-MULTIATOM-VARIANCE-001; atomless probe); for 0 < eta < 1 the roots c_b/(1 -+ sqrt(1 - eta)) with forced upper weight eta/(2(1 + sqrt(1 - eta))) give explicit exact sigma^2 = 0 laws (eta = 3/4), in which eta_hat >= eta for every sample. (H2): the retention reading requires the evaluation law to be P_theta0; under another law the interval estimates an uncentred moment ratio. Numerically: closed-form cell boundaries (quadratic roots of the classifier logit) and Gaussian-CDF cell moments reproduce the recorded door3 references to rounding and the researcher's artifact to <= 2e-15; fresh-seed coverage (4000 replicates) replicates O6.7; the rung n_per_class = 300 under-covers to 0.896 at n = 100; in both sigma^2 = 0 laws the Wald interval is conservative, not liberal. Literature: the method is textbook prior art; no direct statement of the fixed-partition uncentred correlation-ratio CLT with this influence function was located (search gap, not novelty).

Assumptions

source frozen at score-oracle-robustness@a5f905e (study handoff b9a42c3); the researcher's instrument, transcript and plan file were not read; the researcher's artifacts and literature audit were opened only after the audit's own numbers and table existed, as comparison material
the independent instrument py/audit_score_oracle_retention_uncertainty.py does not import, extend or copy py/score_oracle_retention_uncertainty.py; claim-level identities are fractions.Fraction; population references are closed-form Gaussian-CDF differences plus two independent quadratures; Monte Carlo seeds are fresh (SeedSequence 20260906)
the imported hypotheses of FI-QUANT-IDENTITY and PROXY-TRUE-RETAINED-FI were rechecked for the frozen map q o s_hat, not re-proved
scope is O6.1-O6.5 and the two claim nodes; vector scores, refitted rules, weights, bootstrap comparisons and any public API are flagged, not chased
no src/ or public API change is authorized by this audit

independent publication-grade verification and hardening of O6; authorizes removal of the 'not independently audited' notes, not a library uncertainty API

Where it stops

Verified does not mean deployable: the interval is conditional on a frozen rule, needs an oracle-score evaluation sample from the reference law, has no finite-sample guarantee (0.90-0.92 coverage at n = 100), and is unsupported (conservative, width O(1/n)) when sigma^2 = 0. Nothing here covers vector scores, refitted rules or weights.

Bounded by

CE-O6-ETA-ZERO-MULTIATOM-VARIANCE-001
The influence variance sigma^2 of the frozen-rule scalar retention plug-in vanishes only when S | Z = b is supported on at most two atoms per cell; in particular an atomless (or many-atom) cell of positive probability implies sigma^2 > 0 (O6.4 wording and the A4 parenthetical of RETENTION-PLUGIN-CLT-FROZEN-SCALAR as recorded on 5 September 2026).

Closest prior work

Why each source is not this result: the search records

Nearest sources, checked 5 September 2026

van der Vaart (1998), `vanderVaart-1998`.: Chapter 3 "Delta Method", §3.1 "Basic Result" (p. 25) — primary text (scanned contents, ETH library). "3.1 Theorem" is the delta method with the exact hypotheses O6 uses (map defined on a subset of $\mathbb R^k$ , differentiable at $\theta$ , $T_n$ with values in the domain) — primary text in the author's lecture notes with identical numbering (Kleijn/UvA host), the 1998 book itself corroborated (secondary). The sample variance under $EX^4<\infty$ is Example 3.2, not an unnumbered example — primary text. Chapter 20 "Functional Delta Method" (§20.1 von Mises calculus, §20.2 Hadamard differentiability) — primary text (contents). Theorem 5.23 (M-estimator asymptotic normality) sits in §5.3 "Asymptotic Normality" (pp. 51–60) — section primary text, number secondary. Corrected: Example 3.2 named; Chapter 20 added.
Cramér (1946), `Cramer-1946`.: Chapter 28 "Asymptotic properties of sampling distributions" (p. 363), §28.4 "Functions of moments" — primary text (archive.org scan of the contents). Companion finite-sample algebra: §27.4 "The variance", §27.7 "Functions of moments". Confirmed; the "cited from memory" caveat is lifted.
Serfling (1980), `Serfling-1980`.: Chapter 3 "Transformations of Given Statistics" (p. 117), Chapter 2 "The Basic Sample Statistics" (p. 55) — primary record (Google Books). The labels "Theorem A" / "Theorem B" in §3.3, and which of the two needs only differentiability at the point — unresolved (no readable text reachable: Wiley and the e-book front matter refuse, the archive copy has no OCR). The researcher's "Thm B corroborated by two secondary citations" stands as secondary; the auditor adds no confirmation. Hampel-Ronchetti-Rousseeuw-Stahel-1986.** Chapter 2 "One-Dimensional Estimators" begins p. 78 — primary record; $V(T,F)=\int\mathrm{IF}^2dF$ is cited to p. 85, inside §2.1 — secondary. Sub-section letter and equation numbers — unresolved.
Wishart (1932), `Wishart-1932`.: Biometrika 24(3/4):441–456 — primary record (journal TOC). Normal-theory content — secondary. Chapter 26 "Statistical Relationship: Linear Regression and Correlation", Chapter 27 "Partial and Multiple Correlation" — primary text (front matter of the archive scan). The section treating the correlation ratio's standard error, and whether a non-normal moment-based variance is given — unresolved (body text not retrievable). Not registered; the gap stays in gaps.md.

The full search record

Relationships

Evidence references

Reviewed inputs and supporting evidence; these are not proof prerequisites.