Singularity changes the plug-in retention rate

Recorded 6 September 2026 by the RETENTION-PLUGIN-VECTOR session (study study RETENTION-PLUGIN-VECTOR (git history)). Verdict: proved, as a bridge from the vector delta method and the determinant differential; the vector special case of OPEN-RETENTION-UNCERTAINTY it settles is stated exactly below and OPEN-RETENTION-UNCERTAINTY stays open for refitted rules, weights and the no-oracle case. Instrument py/retention_plugin_vector.py; artifacts under artifacts/RETENTION-PLUGIN-VECTOR/. Independently audited 6 September 2026 (AUDITS/AUDIT-RETENTION-PLUGIN-VECTOR-001.md): verified with hardened assumptions — O7.1–O7.3, the Wald statement under (A4), the ellipsoid characterisation and the endpoint rate hold; H1 corrects the $\eta_D=1$ sentence (singular $\hat V$ samples, CE-O7-UNIT-RETENTION-SINGULAR-SAMPLE-001), H2 reduces the Wald "iff" to "if", H3 marks the endpoint-coverage convergence as measured, H4 names the degenerate ellipsoid.

Cite versus derive. The literature-first pass of 5 September 2026 (LITERATURE/audits/RETENTION-PLUGIN-CLT-FROZEN-VECTOR-5-September-2026.md) found the method published: the vector delta method (van der Vaart 1998, Thm 3.1), the determinant and inverse differentials (Magnus & Neudecker §8.3–8.4), influence functions of $\Lambda$-type and canonical-correlation parameters (Radhakrishnan & Kshirsagar 1981; Romanazzi 1992) and asymptotic normality of eigenvalue functionals under finite fourth moments, repeated eigenvalues allowed (Muirhead & Waternaux 1980; Fang & Krishnaiah 1982). None of it is re-proved here. What O7 adds is project algebra inside that method: the uncentred fixed-partition form on the cell moments the library actually computes, the explicit $\psi$ and its $d=1$ reduction, the plug-in variance and its consistency, and the endpoints — where the vector case genuinely differs from O6. A search gap, not novelty.

O7.0 Normalized target (protocol A)

• Level: information_accounting, conditional on a frozen empirical_inductive_quantizer. Criterion: D (the geometric-mean retention INFO-D-EFFICIENCY).
• 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 evaluation law $P$, equally weighted.
• Observables: $S_i=s(X_i)\in\mathbb R^d$ (true score) and $Z_i=q(\hat s(X_i))\in\{1,\dots,K\}$; as in O6 the pairs $(S_i,Z_i)$ are iid because $q\circ\hat s$ is a fixed measurable map.
• Population target: $$p_b=P(Z=b),\quad m_b=E[S\mathbf 1_{Z=b}],\quad c_b=\frac{m_b}{p_b},\quad V=E[SS^\top],\quad I_Z=\sum_b\frac{m_bm_b^\top}{p_b},\quad \eta_D=\Big(\frac{\det I_Z}{\det V}\Big)^{1/d}.$$ Under a regular model at the reference law ($E[S]=0$), $V=I_{\rm full}$ and $I_Z=\operatorname{Var}(E[s\mid q(\hat s)])$ (U1 with $q$ fixed in $\theta$, FI-QUANT-IDENTITY; O4, PROXY-TRUE-RETAINED-FI), so $\eta_D=(\det R)^{1/d}$ with $R=V^{-1/2}I_ZV^{-1/2}$ is exactly the true geometric-mean retention I1 (INFO-D-EFFICIENCY). Both $\eta_D$ and the influence function below are invariant under $S\mapsto AS$, $A$ nonsingular (D-REPARAM-INVARIANCE), so nothing here depends on the parametrisation of $\theta$.
• Classical reading (stated once). With $M=E[S\mathbf 1_Z^\top]$ and $P=\operatorname{diag}(p)$, $I_Z=MP^{-1}M^\top$ and $\det I_Z/\det V=\det(V^{-1}MP^{-1}M^\top)=\prod_{i\le d}\rho_i^2$: $\eta_D$ is the geometric mean of the squared uncentred canonical correlations between $S$ and the cell indicator $\mathbf 1_Z$. At the reference law the uncentred and centred objects coincide, $I_Z$ is the between-cell matrix, $V$ the total matrix, and $\prod_i(1-\rho_i^2)=\det(V-I_Z)/\det V$ is Wilks' $\Lambda$ for the grouping $Z$. O7 is therefore the fixed-alternative first-order theory of a MANOVA determinant ratio under finite fourth moments and no normality, with the grouping a fixed map of a different variable.
• Estimator: the plug-in on the same evaluation sample, $\hat p_b=n_b/n$, $\hat m_b=n^{-1}\sum_iS_i\mathbf 1_{Z_i=b}$, $\hat V=n^{-1}\sum_iS_iS_i^\top$, $\hat I_Z=\sum_{b:\,n_b>0}\hat m_b\hat m_b^\top/\hat p_b$, $\hat\eta_D=(\det\hat I_Z/\det\hat V)^{1/d}$ with $0/0:=0$ on empty cells and $\hat\eta_D:=0$ when $\det\hat V=0$. Scores are never centred.
• Excluded (study scope): importance weights, growing $K$, rules refitted on the evaluation sample, $D_s$/profiled retention, classifier calibration, bootstrap comparisons, any public uncertainty API.

Assumptions. (A1) $p_b>0$ for every $b$; (A2) $E\|S\|^4<\infty$; (A3) $V\succ0$; (A3′) $I_Z\succ0$; (A4) $\sigma^2>0$ with $\sigma^2$ defined in O7.2. (A2) is automatic for bounded scores (mixture-fraction scores at an interior reference point); (A3) is the library's nonsingular-information hypothesis; (A3′) is new to the vector case — by U4 (FI-RANK-CEILING) it needs $K\ge d+1$ at the reference law, and it fails exactly at the singular endpoint treated in O7.4(b).

O7.1 Finite-sample identity and library agreement — [BRIDGE]

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

$$\boxed{\; \hat V-\hat I_Z=\frac1n\sum_i\big(S_i-\hat c_{Z_i}\big)\big(S_i-\hat c_{Z_i}\big)^\top\succeq0, \qquad 0\preceq\hat I_Z\preceq\hat V,\qquad 0\le\hat\eta_D\le1. \;}$$

Proof. $\sum_i(S_i-\hat c_{Z_i})(S_i-\hat c_{Z_i})^\top=\sum_iS_iS_i^\top-\sum_bn_b\hat c_b\hat c_b^\top$ because the cross terms in cell $b$ sum to $n_b\hat c_b\hat c_b^\top$, and $n_b\hat c_b\hat c_b^\top=n\,\hat m_b\hat m_b^\top/\hat p_b$. Empty cells contribute to neither side (the $0/0:=0$ convention). $\det$ is monotone on the positive-semidefinite order, so $0\le\det\hat I_Z\le\det\hat V$. This is Kendall & Stuart's decomposition (26.50) in uncentred matrix form. $\square$

Rank. $\operatorname{rank}\hat I_Z\le\min(d,K)$ on any sample, and $\le\min(d,K-1)$ on an exactly centred sample ($\sum_b\hat m_b=0$); at the reference law $\operatorname{rank}I_Z\le\min(d,K-1)$ (U4). So $K\le d-1$ gives $\hat\eta_D=0$ exactly, while $K=d$ gives a population value $0$ with a generically positive plug-in — O7.4(b).

Library agreement. information_report(S, Z).geometric_mean_retention whitens by $\hat V^{-1/2}$ on the retained eigen-subspace of $\hat V$ and returns $\exp(\log\det/\mathrm{rank})$ (reports.py, _report_from_fishers). When $\hat V\succ0$ and rank_rtol retains every direction the whitening cancels in the ratio, $\det(\hat V^{-1/2}\hat I_Z\hat V^{-1/2})=\det\hat I_Z/\det\hat V$, and the library number is $\hat\eta_D$ (measured agreement $1.1\cdot10^{-16}$ at $d=2$, $4\cdot10^{-17}$ at $d=3$). Two caveats outside the theorem: (i) when rank_rtol projects a direction out, the library reports the ratio on the retained subspace with exponent $1/\mathrm{rank}$ — a different, discontinuous estimator whose target (A3) excludes; (ii) the $d$-th root amplifies rounding at an exactly singular $\hat I_Z$: the exact plug-in is $0$ but the library returns $\approx\varepsilon^{1/d}$ (measured $1.6\cdot10^{-9}$ at $d=2$ for an exact zero).

O7.2 Conditional asymptotic normality — [BRIDGE]

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

$$\boxed{\; \sqrt n\,(\hat\eta_D-\eta_D)\;\Rightarrow\;N(0,\sigma^2),\qquad \sigma^2=E[\psi(S,Z)^2],\qquad \psi(S,Z)=\frac{\eta_D}{d}\Big[\,2S^\top I_Z^{-1}c_Z-c_Z^\top I_Z^{-1}c_Z-S^\top V^{-1}S\,\Big]. \;}$$

Writing $N_1=2S^\top I_Z^{-1}c_Z-c_Z^\top I_Z^{-1}c_Z$ (the influence of $\log\det I_Z$, $E[N_1]=d$) and $N_2=S^\top V^{-1}S$ (the influence of $\log\det V$, $E[N_2]=d$),

$$\sigma^2=\frac{\eta_D^2}{d^2}\Big[\operatorname{Var}N_1-2\operatorname{Cov}(N_1,N_2)+\operatorname{Var}N_2\Big],$$

the numerator–denominator covariance form. At $d=1$, $I_Z=\eta v$ gives $\psi=(2c_ZS-c_Z^2-\eta S^2)/v=((1-\eta)S^2-(S-c_Z)^2)/v$, exactly O6.2.

Proof. Let $T=(\mathbf 1_{Z=b},\,S\mathbf 1_{Z=b},\,\operatorname{vech}SS^\top)_{b\le K}$ and $\theta=E[T]=(p,M,V)$. By (A2) $T$ has finite second moments, so $\sqrt n(\bar T-\theta)\Rightarrow N(0,\operatorname{Cov}T)$. Define the estimator's own functional on the whole space, $\phi(p,M,V)=\big(\det\sum_{b:p_b>0}m_bm_b^\top/p_b\,/\det V\big)^{1/d}$ with $\phi:=0$ where $\det V=0$ or the determinant ratio is $\le0$; then $\hat\eta_D=\phi(\bar T)$ for every sample, empty cells included. On the open set $\{p_b>0\ \forall b,\ V\succ0,\ I_Z\succ0\}$, which contains $\theta$ by (A1), (A3), (A3′), $\phi$ coincides with the $C^\infty$ function $g=\exp\{d^{-1}[\log\det I_Z-\log\det V]\}$, so $\phi$ is differentiable at $\theta$ and the delta method (van der Vaart Thm 3.1) gives $\sqrt n(\phi(\bar T)-\phi(\theta))\Rightarrow N(0,\operatorname{Var}(\nabla g^\top T))$. The gradient: by $d\log\det A=\operatorname{tr}A^{-1}dA$ (Magnus & Neudecker §8.3) and $dI_Z=\sum_b[(dm_b)m_b^\top+m_b(dm_b)^\top]/p_b-\sum_bm_bm_b^\top dp_b/p_b^2$,

$$d\log\det I_Z=\sum_b\big[2c_b^\top I_Z^{-1}dm_b-c_b^\top I_Z^{-1}c_b\,dp_b\big],\qquad d\log\det V=\operatorname{tr}(V^{-1}dV).$$

Substituting the increments $dp_b=\mathbf 1_{Z=b}-p_b$, $dm_b=S\mathbf 1_{Z=b}-m_b$, $dV=SS^\top-V$ of one observation,

$$\nabla g^\top(T-\theta)=\frac{\eta_D}{d}\Big[2S^\top I_Z^{-1}c_Z-c_Z^\top I_Z^{-1}c_Z-S^\top V^{-1}S -\big(2\textstyle\sum_bm_b^\top I_Z^{-1}c_b-\sum_bp_bc_b^\top I_Z^{-1}c_b-\operatorname{tr}V^{-1}V\big)\Big],$$

and the bracketed constant is $2d-d-d=0$ because $\sum_bm_bc_b^\top=\sum_bp_bc_bc_b^\top=I_Z$. Hence $\nabla g^\top(T-\theta)=\psi$, $E[\psi]=0$ by the same three identities, and $\operatorname{Var}\psi=E[\psi^2]$. $\square$

Canonical-correlation form (a check, not a second proof). With $V$-orthonormal canonical directions $a_i$ of $S$ and $P$-orthonormal directions $b_i$ of $\mathbf 1_Z$ ($Mb_i=\rho_iVa_i$), put $u_i=a_i^\top S$, $v_i=b_{i,Z}$. Then $S^\top V^{-1}S=\sum_iu_i^2$, $c_Z^\top I_Z^{-1}c_Z=\sum_iv_i^2$ and $S^\top I_Z^{-1}c_Z=\sum_iu_iv_i/\rho_i$, so

$$\psi=\frac{\eta_D}{d}\sum_{i\le d}\frac{-\rho_i^2u_i^2+2\rho_iu_iv_i-\rho_i^2v_i^2}{\rho_i^2} =\frac{\eta_D}{d}\sum_i\frac{\operatorname{IF}(\rho_i^2)}{\rho_i^2},$$

the average of the per-coefficient influence functions of Romanazzi (1992) in the form restated by his citers (LITERATURE/audits/…VECTOR-5-September-2026.md, post-derivation check). The per-coefficient pieces need simple $\rho_i$; the determinant route above does not, and the sum is well defined with repeated $\rho_i$ (Fang & Krishnaiah 1982 treat exactly that case).

O7.3 A consistent implementable variance — [BRIDGE]

Let $\hat\psi_i=\frac{\hat\eta_D}{d}\big[2S_i^\top\hat I_Z^{-1}\hat c_{Z_i}-\hat c_{Z_i}^\top\hat I_Z^{-1}\hat c_{Z_i}-S_i^\top\hat V^{-1}S_i\big]$ and $\hat\sigma^2=n^{-1}\sum_i\hat\psi_i^2$. Then $\sum_i\hat\psi_i=0$ exactly whenever $\hat I_Z\succ0$ and $\hat V\succ0$, and under (A1)–(A3′) $\hat\sigma^2\to\sigma^2$ almost surely.

Proof. The three sample identities $\sum_iS_i^\top\hat I_Z^{-1}\hat c_{Z_i}=n\operatorname{tr}(\hat I_Z^{-1}\sum_b\hat m_b\hat c_b^\top)=nd$, $\sum_i\hat c_{Z_i}^\top\hat I_Z^{-1}\hat c_{Z_i}=nd$, $\sum_iS_i^\top\hat V^{-1}S_i=nd$ give the exact zero. $\hat\sigma^2$ is a fixed polynomial in $\hat\eta_D$, the entries of $\hat I_Z^{-1}$, $\hat V^{-1}$ and $\hat c_1,\dots,\hat c_K$, whose coefficients are within-cell empirical moments $n^{-1}\sum_iS_{i,j}S_{i,k}S_{i,l}S_{i,m}\mathbf 1_{Z_i=b}$ of order $\le4$. By (A2) and the strong law every such moment converges a.s.; by (A1) $\hat p_b\to p_b>0$, so $\hat c_b\to c_b$; by (A3), (A3′) $\hat V\to V\succ0$ and $\hat I_Z\to I_Z\succ0$, so the inverses and $\hat\eta_D$ converge; the polynomial is continuous at the limit and its value there is $E[\psi^2]$. Empty cells are a.s. transient by Borel–Cantelli under (A1). $\square$

O7.4 Wald interval and its endpoints — [BRIDGE]

Under (A1)–(A4), $P\big(\eta_D\in\hat\eta_D\pm z_{1-\alpha/2}\hat\sigma/\sqrt n\big)\to1-\alpha$ (O7.2, O7.3, Slutsky). Audit note (H2): this is an "if"; when $\sigma^2=0$ the first-order theory gives no level and the interval is unsupported (measured conservative, O7.7). The endpoints are where the vector case differs from O6.

(a) The $\sigma^2=0$ set is an ellipsoid per cell. Under (A1)–(A3′), $\sigma^2=0$ iff for every cell $b$ the conditional law of $S$ given $Z=b$ is supported on

$$\boxed{\; \mathcal E_b=\Big\{s:\ (s-VI_Z^{-1}c_b)^\top V^{-1}(s-VI_Z^{-1}c_b)=c_b^\top I_Z^{-1}(V-I_Z)I_Z^{-1}c_b\Big\}, \;}$$

an ellipsoid in the $V^{-1}$ metric centred at $VI_Z^{-1}c_b$ (the right-hand side is $\ge0$ because $V\succeq I_Z$; when it is $0$ the ellipsoid is that single point — audit note H4). The identity behind it holds for anisotropic $V$: test_o7_audit_ellipsoid_identity_holds_on_an_anisotropic_rational_law.

Proof. $\psi=0$ a.s. iff $s^\top V^{-1}s-2s^\top I_Z^{-1}c_b+c_b^\top I_Z^{-1}c_b=0$ on the support of each conditional law; completing the square in the $V^{-1}$ metric turns the left side into $(s-VI_Z^{-1}c_b)^\top V^{-1}(s-VI_Z^{-1}c_b)-c_b^\top I_Z^{-1}VI_Z^{-1}c_b+c_b^\top I_Z^{-1}c_b$. $\square$

Consequences.

• Absolutely continuous cells force (A4). If $S\mid Z=b$ has a Lebesgue density on some cell of positive probability, $\sigma^2>0$: $\psi$ is a quadratic polynomial in $s$ with quadratic part $-(\eta_D/d)\,s^\top V^{-1}s\ne0$, so its zero set is Lebesgue-null.
• Atomless is not enough for $d\ge2$ — the naive lift of O6.4's "an atomless cell of positive probability implies $\sigma^2>0$ when $\eta>0$" is false: CE-O7-ELLIPSOID-ZERO-VARIANCE-001. Four cells related by quarter turns, $S\mid Z=0$ uniform on $\{(3,4),(3,-4)\}$, $p_b=1/4$: $E[S]=0$, $V=\tfrac{25}2I$, $I_Z=\tfrac92I$, $\eta_D=9/25$ (det ratio $81/625$), and $\psi=0$ at every atom. The cell-0 ellipsoid is the circle $|s-(25/3,0)|^2=(20/3)^2$ and, as a polynomial, $\psi_r(s)=-(2r/25)\,(s_1^2+s_2^2-\tfrac{50}3s_1+25)$ with $r=\eta_D^2$ — it vanishes on the whole circle. Every law on that circle with mean $(3,0)$ therefore has $\sigma^2=0$ at $\eta_D=9/25$, including the atomless law uniform on the arc of half-angle $\alpha$ about the far point, $\sin\alpha/\alpha=4/5$ ($\alpha\approx1.1311$); measured on it, the Wald interval is conservative (O7.7). At $d=1$ the ellipsoid is the two-point set $\{c_b/(1\mp\sqrt{1-\eta})\}$ of O6.4, which is why O6's atomless remark was true there and only there.
• $\eta_D=1$ iff $V=I_Z$ iff $S=c_Z$ a.s. (zero within-cell scatter); then $\psi\equiv0$, the ellipsoid degenerates to the point $c_b$, and $\hat\eta_D=1$ for every sample with $\hat V\succ0$, i.e. whenever the occupied cells' means span $\mathbb R^d$. Audit hardening H1: a sample confined to fewer spanning cells has $\hat V$ singular, $\hat\eta_D=0$ by the estimator's own convention, and library value $1$ (projection) — CE-O7-UNIT-RETENTION-SINGULAR-SAMPLE-001; the event has probability $\to0$ exponentially under (A1).
• Where (A4) fails the first-order limit is degenerate and the interval is unsupported; measured (O7.7) it is conservative on the arc law with width $O(1/n)$. The quadratic-form limit of $n(\hat\eta_D-\eta_D)$ is not derived.

(b) Singular $I_Z$: the plug-in is biased upward at rate $n^{-(d-r)/d}$. Let (A1), (A3) hold with only $E\|S\|^2<\infty$, and let $r=\operatorname{rank}I_Z<d$ (so $\eta_D=0$), $U\in\mathbb R^{d\times(d-r)}$ an orthonormal basis of the null space of $I_Z$. Then

$$\boxed{\;\hat\eta_D=O_p\big(n^{-(d-r)/d}\big),\;}$$

and if moreover $K\ge d$ and $E[U^\top SS^\top U\,\mathbf 1_{Z=b}]\succ0$ for every cell, $n^{(d-r)/d}\hat\eta_D$ converges in law to a strictly positive random variable. Registered as RETENTION-PLUGIN-SINGULAR-ENDPOINT-RATE.

Proof. Let $R$ span the range of $I_Z$ and write $a_b=R^\top\hat m_b$, $u_b=U^\top\hat m_b$. Since $U^\top m_b=0$ for every $b$ ($I_Z=\sum_bm_bm_b^\top/p_b\succeq m_bm_b^\top/p_b$), $\sqrt n\,u_b\Rightarrow G_b$ jointly, with $G_b\sim N(0,E[U^\top SS^\top U\mathbf 1_{Z=b}])$ and $G_b\perp G_{b'}$ for $b\ne b'$ (the cross-covariance $E[U^\top S\mathbf 1_{Z=b}(U^\top S\mathbf 1_{Z=b'})^\top]-0=0$). In the basis $[R\ U]$, $\hat I_Z$ has blocks $A_n=\sum_ba_ba_b^\top/\hat p_b\to A=R^\top I_ZR\succ0$, $B_n=\sum_ba_bu_b^\top/\hat p_b=O_p(n^{-1/2})$, $C_n=\sum_bu_bu_b^\top/\hat p_b=O_p(n^{-1})$, and $\det\hat I_Z=\det A_n\det S_n$ with the Schur complement $S_n=C_n-B_n^\top A_n^{-1}B_n=\sum_b(u_b-\Gamma_na_b)(u_b-\Gamma_na_b)^\top/\hat p_b$, $\Gamma_n=B_n^\top A_n^{-1}$. By the CLT, the strong law and Slutsky, $nS_n\Rightarrow\sum_bW_bW_b^\top/p_b$ with $W=G(I-\Lambda)$, $\Lambda_{b'b}=a_{b'}^{\infty\top}A^{-1}a_b^\infty/p_{b'}$, $a_b^\infty=R^\top m_b$ — $\Lambda$ is a rank-$r$ projection. Hence $n^{d-r}\hat\eta_D^d=n^{d-r}\det\hat I_Z/\det\hat V\Rightarrow\det A\cdot\det(WP^{-1}W^\top)/\det V$, which is $O_p(1)$: the upper bound. Under the nondegeneracy hypothesis the $G_b$ are independent nondegenerate Gaussians, so $G$ has a Lebesgue density on $\mathbb R^{(d-r)\times K}$ and $W=G(I-\Lambda)$ one on the $(d-r)(K-r)$-dimensional image space; when $K-r\ge d-r$ the rank-deficient matrices form a proper algebraic subvariety of that space, so $\operatorname{rank}W=d-r$ a.s. and $\det(WP^{-1}W^\top)>0$ a.s. $\square$

Consequences. At $d=1$ ($r=0$) this is O6's measured $O(1/n)$ bias. At $d\ge2$ with $r=d-1$ the bias is $n^{-1/d}$, slower than the CLT scale: the plug-in and its Wald half-width $\hat\sigma/\sqrt n$ are then of the same order (measured, O7.7), so coverage of the true value $0$ is measured to stabilise at a law-dependent constant, not at $1-\alpha$; it happened to be $\approx0.95$ on the $d=2$, $K=2$ law and $\approx0.80$ on the $d=3$, $K=3$ law. Audit note (H3): the joint limit of $(n^{(d-r)/d}\hat\eta_D,\,n^{(d-r)/d}\hat\sigma/\sqrt n)$ is not derived, so neither the half-width order nor the coverage convergence is a theorem; the audit confirmed the limit law of $n^{(d-r)/d}\hat\eta_D$ itself against its Gaussian ingredients (closed form $E=2/\pi$ at $d=2$, $K=2$). This is the classical dimensionality problem (Seo, Kanda & Fujikoshi 1995): the right tool at the singular endpoint is a rank test, not this interval. The reference-law instance is $K=d$: U4 forces $\eta_D=0$, the sample plug-in is positive of order $n^{-1/d}$, and the library number is that plug-in.

(c) Other unsupported cases. Empty evaluation cells are handled by $0/0:=0$ and vanish exponentially under (A1); no uniformity over small $p_b$. The interval may leave $[0,1]$ near the endpoints (measured: never, on the regular laws; at the singular endpoint its lower end is negative in most replicates, which is how it covers $0$). No transformation was attempted — one estimator, one interval, per study. No finite-sample guarantee: O7.7 shows how far from first order a heavy-tailed score can be.

O7.5 What the interval does and does not measure — [BRIDGE]

Verbatim from O6.5 with $\eta$ replaced by $\eta_D$: the interval is for the true geometric-mean retention conditional on the frozen rule; the library's self-reported surrogate replaces $S$ by $\hat s$, its population value $\tilde\eta_D$ is a different number (O4), the same theorem gives the surrogate plug-in a valid interval around that wrong target, and the $O(n^{-1/2})$ interval excludes $\tilde\eta_D$ with probability $\to1$ whenever $\tilde\eta_D\ne\eta_D$. Without an oracle-score evaluation sample nothing here applies. Since the D-efficiency is the geometric mean of the retention spectrum, the interval says nothing about the worst retained direction (OPEN-D-DIRECTIONAL-BOUND).

O7.6 Self-adversarial pass (protocol G)

• Ties, duplicate scores: irrelevant; the functional is smooth in moments (selftest carries a duplicate atom).
• Singleton/empty cells: O7.4(c); under (A1) transient.
• Singular information: $V$ singular is excluded by (A3) — the library projects instead (O7.1, a different estimator); $I_Z$ singular is O7.4(b).
• Nuisance singularity, $D_s$: out of scope; the profiled retention is a Schur-complement functional and needs its own influence function.
• Atomic laws: the ellipsoid characterisation; two atoms per cell suffice for $\sigma^2=0$ in every $d$, and in $d\ge2$ atomless singular laws do too.
• Hidden compactness: none used.
• First-order-to-finite jumps: none claimed. Measured: $n\cdot$bias $\approx2$–$3$ on the $d=2$ laws, but still growing at $n=3000$ on the $d=3$ Hermite law ($E\|S\|^4=4259$) — pre-asymptotic, see O7.7.
• Empirical-to-population jumps: none — the rule is frozen; refitted rules are OPEN-RETENTION-UNCERTAINTY's remainder.
• Score-estimation error: enters only through the label map.
• Heavy tails: (A2) is used for the CLT of $SS^\top$ and the strong law of the fourth within-cell moments. It is a real constraint in practice: the polynomial scores of O7.7 satisfy (A2) yet need $n\gg3000$ for nominal coverage, while bounded mixture-fraction scores are nominal at $n=100$.
• Repeated canonical correlations: the determinant route needs none.
• New-event extension: the rule is frozen, so every evaluation event is assigned by $q\circ\hat s$; nothing new.

O7.7 Measured (protocol D, run before the proof was trusted) — [MEASURED]

Exact identities (selftest, fractions.Fraction): on integer samples in $d=2$ ($n=12$, $K=5$ with one declared-empty cell and a duplicate atom) and $d=3$ ($n=14$, $K=4$): the within-scatter identity, $0\le\det$-ratio$\le1$ and $\sum_i\hat\psi_i=0$ hold exactly; the exact Gateaux difference quotients of the rational functional $r=\hat\eta_D^d$ at $\varepsilon=2^{-10},2^{-11},2^{-12}$ approach $\psi_r=d\eta_D^{d-1}\psi$ with a remainder that halves per halving of $\varepsilon$ (ratio within $0.02$ of $2$); the exactly centred $K=d=2$ sample has $\det$-ratio exactly $0$; the $d=1$ reduction agrees with O6's $\psi$ to $1.5\cdot10^{-15}$; library agreement $\le1.1\cdot10^{-16}$.

Population references (popref): interval partitions of $x\sim N(0,1)$ with the Hermite scores $S=(x,x^2-1)$ ($d=2$, cuts $-1,0,1$) and $S=(x,x^2-1,x^3-3x)$ ($d=3$, cuts $\pm1.5,\pm0.5$); every cell moment up to order $4d+2$ is a truncated-Gaussian moment (three-term recursion), so $\eta_D$ and $\sigma^2$ are closed-form; a composite Gauss–Legendre route with pieces aligned to the cuts agrees to $1.5\cdot10^{-14}$ ($d=2$) and $1.0\cdot10^{-13}$ ($d=3$); $E[S]=0$, $E[\psi]\le1.3\cdot10^{-16}$, $E\|S\|^4=83$ and $4259$ (the hand value $3+20+60=83$ checks). A three-component Gaussian location mixture with two interior fractions ($0.35,0.25$; bounded scores, $|s|\le2.5$, $E\|S\|^4=22.2$; cuts $-2,-0.5,1$) by two aligned quadrature routes agreeing to $2.3\cdot10^{-15}$. A library-fitted rule: optimize_partition (D-exchange, $K=4$) on 4000 training draws of the $d=2$ Hermite score, compiled to its Mahalanobis rule; its cells in $x$ are three intervals located by bisection (cuts $-1.3918,\,0.0138,\,1.3753$), and the closed-form moments per interval agree with a quadrature that asks the compiled rule for every node's label to $5.3\cdot10^{-15}$.

law	$d$	$K$	$p_b$	$\eta_D$	$\sigma$
Hermite $d=2$	2	4	0.159, 0.341, 0.341, 0.159	0.690666	0.673175
Hermite $d=3$	3	5	0.067, 0.242, 0.383, 0.242, 0.067	0.629969	1.624099
bounded mixture	2	4	0.178, 0.305, 0.254, 0.263	0.330317	0.621795
library rule on Hermite $d=2$	2	4	0.424, 0.085, 0.410, 0.082	0.743363	0.557001
Hermite $d=2$, $K=2$ (cut $0$)	2	2	0.5, 0.5	0 (rank 1)	—
Hermite $d=3$, $K=3$ (cuts $\pm0.6$)	3	3	0.274, 0.451, 0.274	0 (rank 2)	—
arc law (fixture, atomless)	2	4	1/4 each	9/25	0

Coverage of the 95% Wald interval, 2000 independent samples per size, SeedSequence([20260906, law, n]):

law	$n$	coverage ± SE	SD$(\hat\eta_D)$	$\sigma/\sqrt n$	mean $\hat\sigma/\sqrt n$	rel. RMSE $\hat\sigma/\sigma$	$n\cdot$bias	plug-in skew	oracle SD / skew
bounded mixture	100	0.950 ± 0.005	0.0627	0.0622	0.0675	0.168	2.80	+0.16	1.01 / +0.20
	300	0.945 ± 0.005	0.0363	0.0359	0.0370	0.083	2.46	−0.12	1.01 / −0.02
	1000	0.956 ± 0.005	0.0191	0.0197	0.0198	0.040	1.70	+0.00	0.97 / +0.07
	3000	0.945 ± 0.005	0.0116	0.0114	0.0114	0.022	2.51	−0.02	1.02 / +0.01
Hermite $d=2$	100	0.681 ± 0.010	0.0554	0.0673	0.0366	0.480	3.16	+0.60	0.82 / −0.52
	300	0.813 ± 0.009	0.0342	0.0389	0.0284	0.362	3.80	+0.63	0.88 / −0.43
	1000	0.892 ± 0.007	0.0204	0.0213	0.0186	0.278	2.86	+0.59	0.96 / −0.26
	3000	0.937 ± 0.005	0.0118	0.0123	0.0117	0.198	2.90	+0.49	0.96 / −0.14
library rule	100	0.816 ± 0.009	0.0405	0.0557	0.0343	0.403	2.01	+0.04	0.73 / −0.63
	300	0.861 ± 0.008	0.0276	0.0322	0.0233	0.346	2.03	+0.30	0.86 / −0.63
	1000	0.889 ± 0.007	0.0166	0.0176	0.0150	0.295	2.70	+0.50	0.94 / −0.47
	3000	0.927 ± 0.006	0.0097	0.0102	0.0095	0.242	2.80	+0.51	0.96 / −0.25
Hermite $d=3$	100	0.399 ± 0.011	0.0706	0.1624	0.0394	0.761	8.9	+0.69	0.43 / −0.55
	300	0.556 ± 0.011	0.0579	0.0938	0.0329	0.656	12.6	+0.43	0.62 / −0.52
	1000	0.733 ± 0.010	0.0414	0.0514	0.0290	0.480	15.0	+0.82	0.81 / −0.50
	3000	0.827 ± 0.008	0.0272	0.0297	0.0222	0.362	19.3	+0.83	0.92 / −0.45

Reading. On the bounded mixture scores — the library's mixture-fraction use case — the interval is nominal from $n=100$: the spread of $\hat\eta_D$ equals $\sigma/\sqrt n$, $\hat\sigma$ is within 8% of $\sigma$ at $n=100$ and within 3% beyond, and the studentized statistic has no skew. On the Hermite scores every first-order quantity converges as the theorem says (SD$\to\sigma/\sqrt n$, $\hat\sigma\to\sigma$, oracle-studentized SD $\to1$) but slowly: $\hat\sigma$ under-estimates $\sigma$ by 46% at $n=100$ and 5% at $n=3000$ in $d=2$, by 76% and 25% in $d=3$, because $\hat\sigma^2$ is a sample eighth ($d=2$) or twelfth ($d=3$) moment of $x$. The interval is therefore liberal there at every size tested — a second-order, heavy-tail effect governed by $E\|S\|^4$ (83 and 4259 against 22 for the mixture), not a counterexample to O7.2–O7.4. Bias is $O(1/n)$ in $d=2$ and still pre-asymptotic in $d=3$ at $n=3000$. No regular-law interval left $[0,1]$; empty cells occurred in 4 of 32,000 replicates (all at $n=100$, in cells with $p_b\le0.085$).

Endpoints (same seeds):

law	$n$	covers $0$	mean $\hat\eta_D$	$n^{1/d}\,$mean $\hat\eta_D$	median half-width / $\hat\eta_D$
Hermite $d=2$, $K=2$ ($r=1$)	100	0.926	0.0704	0.704	2.77
	300	0.946	0.0379	0.656	2.90
	1000	0.954	0.0199	0.631	3.03
	3000	0.956	0.0114	0.627	2.95
Hermite $d=3$, $K=3$ ($r=2$)	100	0.720	0.1319	0.612	1.65
	300	0.753	0.0853	0.571	1.82
	1000	0.788	0.0546	0.546	1.81
	3000	0.802	0.0365	0.526	1.94

$n^{1/d}\hat\eta_D$ stabilises ($0.63$ at $d=2$; $0.53$ and still drifting at $d=3$), the half-width stays a fixed multiple of $\hat\eta_D$, and coverage of the true $0$ tends to a constant — $0.95$ by coincidence at $d=2$, $0.80$ at $d=3$ — exactly the behaviour O7.4(b) predicts. On the arc law ($\sigma^2=0$, $\eta_D=9/25$): coverage $0.996$, $0.999$, $0.999$, $0.999$; $n\cdot$bias $\approx1.4$ and $n\cdot$SD $\approx1.55$ at every size (width $O(1/n)$); $\hat\eta_D\ge\eta_D$ in 85% of replicates — conservative, never liberal, but unlike O6's two-atom law not one-sided.

None of this is a proof of O7.2–O7.4; nothing is claimed about coverage on heavy-tailed scores at the sizes shown beyond the numbers themselves.

O7.8 Information-loss implication and verdict (protocol H)

O7 bounds nothing; it equips the library's reported number, the true geometric-mean D-retention $\eta_D$ of a frozen rule, with a conditional $n^{-1/2}$ error bar on held-out oracle-score data, computable in one extra pass from the same cell moments. Verdict: proved, as a bridge — the CLT is the vector delta method and the determinant differential, the influence function is a sum of published canonical-correlation influence functions, and what is project-level is the uncentred fixed-partition form on cell moments, the plug-in variance with its consistency, the ellipsoid characterisation of $\sigma^2=0$ with its atomless counterexample, and the $n^{-(d-r)/d}$ rate at the singular endpoint. Practical reading: for bounded (mixture-fraction) scores the interval is usable at $n$ in the hundreds; for unbounded polynomial-type scores it is liberal until $n$ is far larger than the fourth moment suggests, and at $K\le d$ or any rank-deficient $I_Z$ it must not be used at all — the plug-in is then biased upward at a rate that is slower than $n^{-1/2}$ for $d\ge2$. Left open and rerouted to OPEN-RETENTION-UNCERTAINTY: rules refitted on the evaluation sample (the boundary non-smoothness OPEN-RETENTION-UNCERTAINTY names); weighted samples; the no-oracle case; the degenerate limits at $\sigma^2=0$ and at the singular endpoint; and the profiled $D_s$ retention.

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