A frozen scalar rule's retention has a limiting distribution

Recorded 5 September 2026 by the SCORE-ORACLE-ROBUSTNESS session (study study SCORE-ORACLE-ROBUSTNESS (git history)). Verdict: proved, as a direct corollary of the delta method; the special case of OPEN-RETENTION-UNCERTAINTY it settles is stated exactly below and the broad claim OPEN-RETENTION-UNCERTAINTY stays open. Instrument py/score_oracle_retention_uncertainty.py; artifacts under artifacts/SCORE-ORACLE-ROBUSTNESS/. Independently audited 5 September 2026 (AUDITS/AUDIT-SCORE-ORACLE-ROBUSTNESS-001.md, instrument py/audit_score_oracle_retention_uncertainty.py): O6.1–O6.3 and the Wald statement of O6.4 verified; the $\sigma^2=0$ characterisation hardened at $\eta=0$ (see the audit notes in O6.4); population references and coverage replicated with fresh seeds.

O6.0 Normalized target (protocol A)

• Level: information_accounting, conditional on a frozen empirical_inductive_quantizer. Criterion: the scalar retention ratio (for one score coordinate the D-, A- and E-efficiencies coincide with it).
• Frozen: training data, provider $\hat s$, rule $q$, reference point $\theta_0$, finite $K$. Random: an independent evaluation sample $X_1,\dots,X_n$ iid from the reference law $P$, equally weighted.
• Observables: $S_i=s(X_i)$ (true scalar score) and $Z_i=q(\hat s(X_i))\in\{1,\dots,K\}$. Because $q\circ\hat s$ is a fixed measurable map, the pairs $(S_i,Z_i)$ are iid with a fixed joint law: the boundary non-smoothness of OPEN-RETENTION-UNCERTAINTY never enters this special case.
• Population target: $$p_b=P(Z=b),\qquad m_b=E[S\,\mathbf 1_{Z=b}],\qquad v=E[S^2],\qquad \eta=\frac{\sum_b m_b^2/p_b}{v},\qquad c_b=\frac{m_b}{p_b}.$$ Under a regular model with $E[S]=0$, $v=I_{\rm full}$ and $\sum_b m_b^2/p_b=I_Z$ (U1, FI-QUANT-IDENTITY, with $q$ fixed in $\theta$), so $\eta=\operatorname{Var}(E[s\mid q(\hat s)])/\operatorname{Var}(s)$ is exactly the true retained fraction of O4 (PROXY-TRUE-RETAINED-FI).
• Estimator: the ordinary plug-in ratio on the same evaluation sample, $\hat p_b=n_b/n$, $\hat m_b=n^{-1}\sum_i S_i\mathbf 1_{Z_i=b}$, $\hat v=n^{-1}\sum_i S_i^2$, $\hat\eta=\sum_b\hat m_b^2/\hat p_b\,/\,\hat v$, with $0/0:=0$ for an empty cell. Scores are never centred.
• Excluded (study scope): importance weights, growing $K$, refitting the rule on the evaluation sample, boundary stability, $D_s$, classifier calibration theory, bootstrap comparisons, any public uncertainty API.

Assumptions. (A1) $p_b>0$ for every $b$; (A2) $E[S^4]<\infty$; (A3) $v=E[S^2]>0$; (A4) $\sigma^2>0$ with $\sigma^2$ defined in O6.2. (A2) is automatic for bounded scores, e.g. every mixture-fraction score at an interior reference point ($|s|\le\max_k 1/\theta_{0k}$); (A3) is the nonsingular unbinned-information hypothesis the library already imposes.

O6.1 Finite-sample identity — [BRIDGE]

For every sample, with $\hat c_b=\hat m_b/\hat p_b$ on nonempty cells,

$$\boxed{\; \hat\eta \;=\; 1-\frac{\sum_i\big(S_i-\hat c_{Z_i}\big)^2}{\sum_i S_i^2} \;}$$

so $\hat\eta$ is one minus the within-cell residual sum of squares about the cell means over the uncentred total sum of squares, and $0\le\hat\eta\le1$ always. Empty cells contribute nothing to either side, which is what the $0/0:=0$ convention encodes. For scalar scores $\hat\eta$ coincides with the public information_report(...).geometric_mean_retention (the whitening divides by $\hat v$; selftest agreement $5\cdot10^{-17}$).

Proof. $\sum_i(S_i-\hat c_{Z_i})^2=\sum_b\big[\sum_{i\in b}S_i^2-n_b\hat c_b^2\big] =\sum_iS_i^2-\sum_b n_b\hat c_b^2$ and $n_b\hat c_b^2=(\sum_{i\in b}S_i)^2/n_b =n\,\hat m_b^2/\hat p_b$. Divide by $\sum_iS_i^2=n\hat v$. $\square$

O6.2 Conditional asymptotic normality — [BRIDGE]

Under (A1)–(A3), conditionally on the frozen rule,

$$\boxed{\; \sqrt n\,(\hat\eta-\eta)\;\Rightarrow\;N(0,\sigma^2),\qquad \sigma^2=E[\psi(S,Z)^2],\qquad \psi(S,Z)=\frac{(1-\eta)\,S^2-(S-c_Z)^2}{v}. \;}$$

Equivalently, writing the numerator influence $N_1=2c_ZS-c_Z^2$ (so that $E[N_1]=\eta v$) and the denominator influence $S^2$,

$$\sigma^2=\frac{\operatorname{Var}(N_1)-2\eta\operatorname{Cov}(N_1,S^2)+\eta^2\operatorname{Var}(S^2)}{v^2},$$

which is the numerator–denominator covariance form the study asked for.

Proof. Let $T_i=(\mathbf 1_{Z_i=b},\,S_i\mathbf 1_{Z_i=b},\,S_i^2)_{b=1..K}\in\mathbb R^{2K+1}$, $\theta=E[T_1]=(p_b,m_b,v)_b$. By (A2) $T_1$ has finite second moments (the largest is $E[S^4]$), so the multivariate CLT gives $\sqrt n(\bar T-\theta)\Rightarrow N(0,\Sigma)$, $\Sigma=\operatorname{Cov}(T_1)$. The map $g(p,m,v)=\sum_b m_b^2/p_b\,/\,v$ is $C^\infty$ on $\{p_b>0\ \forall b,\ v>0\}$, which contains $\theta$ by (A1), (A3). The delta method (van der Vaart 1998, Thm 3.1) yields $\sqrt n(g(\bar T)-g(\theta))\Rightarrow N(0,\nabla g^\top\Sigma\nabla g)$, and $\nabla g^\top\Sigma\nabla g=\operatorname{Var}(\nabla g^\top T_1)$. With

$$\frac{\partial g}{\partial p_b}=-\frac{m_b^2}{p_b^2v},\qquad \frac{\partial g}{\partial m_b}=\frac{2m_b}{p_bv},\qquad \frac{\partial g}{\partial v}=-\frac{\eta}{v},$$ $$\nabla g^\top(T_1-\theta) =\sum_b\Big[-\frac{m_b^2}{p_b^2v}(\mathbf 1_{Z=b}-p_b)+\frac{2m_b}{p_bv}(S\mathbf 1_{Z=b}-m_b)\Big]-\frac{\eta}{v}(S^2-v) =\frac{2c_ZS-c_Z^2-\eta S^2}{v}+(\eta-2\eta+\eta)=\psi(S,Z),$$

and $2c_ZS-c_Z^2=S^2-(S-c_Z)^2$ gives the boxed form; $E[\psi]=0$ is the vanishing of the constant terms, so $\operatorname{Var}(\psi)=E[\psi^2]$. Finally $\hat\eta=g(\bar T)$ on the event that every cell is nonempty, whose complement has probability at most $\sum_b(1-p_b)^n\to0$ by (A1); hence $\hat\eta$ and $g(\bar T)$ share the limit law. $\square$

O6.3 A consistent implementable variance — [BRIDGE]

Let $\hat\psi_i=\big((1-\hat\eta)S_i^2-(S_i-\hat c_{Z_i})^2\big)/\hat v$ and $\hat\sigma^2=n^{-1}\sum_i\hat\psi_i^2$. Then $\sum_i\hat\psi_i=0$ exactly (by O6.1), and under (A1)–(A3) $\hat\sigma^2\to\sigma^2$ almost surely.

Proof. Expanding, $n^{-1}\sum_i\hat\psi_i^2$ is a fixed polynomial in $(\hat\eta,\hat v^{-1},\hat c_1,\dots,\hat c_K)$ whose coefficients are the within-cell empirical moments $\hat M_{b,k}=n^{-1}\sum_iS_i^k\mathbf 1_{Z_i=b}$, $k=0,\dots,4$. By (A2) and the strong law each $\hat M_{b,k}\to M_{b,k}=E[S^k\mathbf 1_{Z=b}]$ a.s.; by (A1) $\hat p_b=\hat M_{b,0}\to p_b>0$ so $\hat c_b\to c_b$; by (A3) $\hat v\to v>0$; hence $\hat\eta\to\eta$. The polynomial is continuous at the limit point, and its value there is $E[\psi^2]$ expanded in the same moments. $\square$

O6.4 Wald interval — [BRIDGE]

Under (A1)–(A4), $P\big(\eta\in\hat\eta\pm z_{1-\alpha/2}\,\hat\sigma/\sqrt n\big)\to1-\alpha$ (O6.2, O6.3 and Slutsky). Unsupported cases and degeneracies:

• (A4) fails iff $\psi=0$ a.s., i.e. iff for every $b$ with $p_b>0$ the conditional law of $S$ given $Z=b$ is supported on the roots of $(s-c_b)^2=(1-\eta)s^2$. [Audit 5 Sep 2026, hardened.] For $0<\eta<1$ the roots are $s_\pm=c_b/(1\mp\sqrt{1-\eta})$, so at most two atoms per cell (one, $s=0$, when $c_b=0$); the mean constraint fixes the upper-root weight $w=\eta/(2(1+\sqrt{1-\eta}))$ and forces $E[S^2\mid Z=b]=c_b^2/\eta$, and such laws exist (cells $\{2/3,2\}$ and $\{-2/3,-2\}$ with weights $(3/4,1/4)$, $p_b=1/2$: $\eta=3/4$, $E[S]=0$, $\sigma^2=0$). At $\eta=1$ the single root is $s=c_b$ ($S=c_Z$ a.s.). At $\eta=0$ every $c_b=0$ and the equation is the identity $s^2=s^2$: every law with vanishing cell means has $\psi\equiv0$, with any number of atoms or none (CE-O6-ETA-ZERO-MULTIATOM-VARIANCE-001; the atomless probe in the audit's coverage.json). Hence: an atomless cell of positive probability implies (A4) iff $\eta>0$; the original sentence omitted the qualifier. Where (A4) fails the first-order limit is degenerate, $n(\hat\eta-\eta)$ has a non-normal quadratic-form limit (not derived here), and the Wald interval is not supported: measured, it is conservative there (coverage $\to1$ with width $O(1/n)$, $\hat\eta\ge\eta$ exactly in the two-atom law), not liberal.
• Empty evaluation cells: handled by $0/0:=0$; their probability vanishes exponentially under (A1). No uniformity over small $p_b$ is claimed: a cell with $p_b$ of order $1/n$ is outside the theorem.
• Edges: the interval may leave $[0,1]$ near $\eta\in\{0,1\}$; no transformation was attempted (one estimator, one interval, per study).
• No finite-sample guarantee: coverage below is evidence of the first-order statement only.

O6.5 What the interval does and does not measure — [BRIDGE]

The interval is for the true retention $\eta$, conditional on the frozen rule. The library's self-reported surrogate replaces $S$ by $\hat s$; its population value $\tilde\eta=\sum_b\tilde m_b^2/p_b/\tilde v$ with $\tilde m_b=E[\hat s\mathbf 1_{Z=b}]$, $\tilde v=E[\hat s^2]$ is a different number (O4). The same theorem applies verbatim to the surrogate plug-in as an estimator of $\tilde\eta$, so the surrogate carries a valid interval around the wrong target. The gap $\tilde\eta-\eta$ is bias of the proxy, of order one in $n$, and the $O(n^{-1/2})$ interval excludes $\tilde\eta$ with probability $\to1$: evaluation uncertainty and proxy discrepancy separate cleanly once an oracle-score evaluation sample exists. Without an oracle sample nothing here applies (OPEN-RETENTION-UNCERTAINTY's remainder, and OPEN-SCORE-PERTURBATION/OPEN-CLASSIFIER-CALIBRATION-FI).

O6.6 Self-adversarial pass (protocol G)

• Ties, duplicate scores: irrelevant; the functional is smooth in moments.
• Singleton/empty cells: O6.4; under (A1) transient.
• Singular information: excluded by (A3); with $v=0$ the target is undefined, matching the library's refusal.
• Nuisance singularity, $D_s$: out of scope (scalar, no nuisance).
• Atomic laws: the only route to (A4) failure with $0<\eta<1$; recorded above. [Audit: at $\eta=0$ atomless laws also fail (A4); see O6.4.]
• Hidden compactness: none used.
• First-order-to-finite jumps: none claimed; measured $O(1/n)$ bias below.
• Empirical-to-population jumps: none — the rule is frozen, the sample is independent of it; a rule refitted on the evaluation sample voids the iid structure and is exactly what OPEN-RETENTION-UNCERTAINTY still owes.
• Score-estimation error: enters only through the label map and the target's definition; the theorem is conditional on $\hat s$.
• Heavy tails: (A2) is used for the CLT of $S^2$ and the SLLN of fourth within-cell moments; with only $E[S^2]<\infty$ the denominator CLT fails.

O6.7 Measured (protocol D, run before the proof was trusted) — [MEASURED]

Frozen rule: door3 rung n_per_class = 15 (classifier seed 101, four cells, D-exchange seed 7, the rung with the largest published proxy gap). The instrument reproduces the published ladder on door3's own test sample (surrogate 0.9658, true 0.8847). Population references by two independent composite Gauss–Legendre routes (disagreement $1.1\cdot10^{-16}$, tail truncation $<10^{-18}$, closed-form score vs the exact provider $9\cdot10^{-16}$); $E[S]=-4.5\cdot10^{-17}$ confirms the zero-mean score.

quantity	value
$\eta$ (true retention)	0.893663
$\sigma$	0.235410
$\tilde\eta$ (proxy population value)	0.967064
proxy gap $\tilde\eta-\eta$	0.073402
cell probabilities	0.4953, 0.1215, 0.2193, 0.1638
$E[S^4]$	2.7336

Coverage of the 95% Wald interval, 2000 independent evaluation samples per size, SeedSequence(20260905):

$n$	coverage ± SE	SD$(\hat\eta)$	$\sigma/\sqrt n$	mean $\hat\sigma/\sqrt n$	rel. RMSE $\hat\sigma/\sigma$	$n\cdot$bias	studentized skew	covers $\tilde\eta$
100	0.913 ± 0.006	0.02335	0.02354	0.02255	0.167	0.33	+1.06	0.046
300	0.934 ± 0.006	0.01356	0.01359	0.01342	0.088	0.35	+0.20	0.000
1000	0.953 ± 0.005	0.00738	0.00744	0.00742	0.049	0.18	+0.11	0.000
3000	0.949 ± 0.005	0.00433	0.00430	0.00429	0.029	0.40	+0.12	0.000

Reading: the sampling spread matches $\sigma/\sqrt n$ at every size and $\hat\sigma$ converges to $\sigma$; coverage reaches nominal by $n=1000$. The $n=100$ shortfall is a second-order effect — the statistic studentized with the population $\sigma$ has skew $-0.30$ and SD 0.99, while the plug-in studentized statistic has skew $+1.06$: $\hat\sigma$ is small precisely when $\hat\eta$ is high. The measured bias is $O(1/n)$ ($n\cdot$bias $\approx0.3$). No interval left $[0,1]$; no evaluation cell was empty. The interval for $\eta$ essentially never covers the proxy value $\tilde\eta$, as O6.5 predicts. None of this is a proof of O6.2–O6.4, and nothing is claimed about coverage below $n=1000$ beyond the numbers shown.

O6.8 Information-loss implication and verdict (protocol H)

O6 bounds nothing; it equips the true scalar retention $\eta$ (equivalently the D-, A- or E-efficiency of a one-parameter model) with a conditional $n^{-1/2}$ error bar on held-out oracle-score data. Verdict: proved, as a bridge from the delta method, with the following left open and rerouted to OPEN-RETENTION-UNCERTAINTY: vector scores and the geometric-mean retention $(\det R)^{1/d}$ (a smooth matrix functional of the same moments — the matrix influence function is the missing derivation); refitted rules, where the evaluation sample enters the boundaries; weighted samples; and any statement without an oracle score.