A proxy score misreports retention by a computable two-term budget

Recorded 8 September 2026 by the SCORE-ERROR-BUDGET session (study SCORE-ERROR-BUDGET, closure step 3 of OPEN_PROBLEMS.md; git history). Verdict: proved at the first order asked for — the gap between the retention the library reports for a proxy score and the true retention of the same labels is an exact two-term expansion with an explicit curvature remainder (O8.2), the rule-transfer loss is a Markov-plus-margin bound composed with an exact Loewner sandwich (O8.3), and the classifier-calibration chain is one Lipschitz inequality plus the Brier identity (O8.4). Instrument py/score_error_budget.py; artifacts under artifacts/SCORE-ERROR-BUDGET/. Independently audited and corrected on 8 September 2026: AUDITS/AUDIT-SCORE-ERROR-BUDGET-001.md. The core budget survives; the alignment-bound ordering, linear-reduction domain, and classifier lower-bound/deployment readings required correction.

Cite versus derive. The independent five-source triangulation is in LITERATURE/audits/SCORE-ERROR-RETENTION-BUDGET-2026-09-08.md. Boyd–Vandenberghe supplies log-det concavity; Audibert–Tsybakov supplies the margin/Markov comparison method; Bröcker supplies the proper-score decomposition; Barnes–Han–Özgür and Cranmer–Pavez–Louppe supply the information and classifier reductions. The specific retention expansion has a search gap, not a novelty claim.

O8.0 Normalized target (protocol A)

• Level: information_accounting, conditional on a frozen empirical_inductive_quantizer. Criterion: D (the geometric-mean retention, INFO-D-EFFICIENCY) for O8.2; every direction (INFO-RETENTION-SPECTRUM) for O8.1; the trace and the determinant for O8.3.
• Frozen: provider $\hat s$, rule $q$ on score space, reference point $\theta_0$, $K$. No sampling limit is taken. Every statement is an inequality between functionals of one law $P$ on $x$-space — population or empirical. For the empirical law of a sample on which both $s$ and $\hat s$ exist the inequalities hold sample-wise, exactly; combined with O7 the reported number on a held-out sample is off by a sampling term (O7) plus a bias term (here), controlled separately.
• Objects. True score $s\in\mathbb R^d$, proxy $\hat s=s+e$, $V=E[ss^\top]\succ0$, $\tilde V=E[\hat s\hat s^\top]$. Labels $Z=q(\hat s)$ with $p_b=P(Z=b)>0$. Cell means $c_b=E[s\mid Z=b]$, $\tilde c_b=E[\hat s\mid Z=b]=c_b+e_b$, $e_b=E[e\mid Z=b]$. True retained information $I_Z=\sum_bp_bc_bc_b^\top$; reported $\tilde I_Z=\sum_bp_b\tilde c_b\tilde c_b^\top$. Error matrices $\mathcal E=E[ee^\top]$, $\mathcal E_Z=\sum_bp_be_be_b^\top=E[e_Ze_Z^\top]$.
• Scales. The Fisher-whitened score error and its cell-averaged and retention-weighted versions, $$\varepsilon^2=E[e^\top V^{-1}e]=\operatorname{tr}(V^{-1}\mathcal E),\qquad \varepsilon_Z^2=\operatorname{tr}(V^{-1}\mathcal E_Z),\qquad \varepsilon_R^2=\operatorname{tr}(I_Z^{-1}\mathcal E_Z)=E[e_Z^\top I_Z^{-1}e_Z].$$ $R=V^{-1/2}I_ZV^{-1/2}$ with eigenvalues $1\ge\rho_1\ge\dots\ge\rho_d=\rho_{\min}\ge0$ (U7); $\eta_D=(\det R)^{1/d}$; $d-\operatorname{tr}R$ is the rule's trace information loss. All of $\varepsilon,\varepsilon_Z,\varepsilon_R,\rho_i,\eta_D$ are invariant under $(s,\hat s)\mapsto(As,A\hat s)$, $A$ nonsingular (D-REPARAM-INVARIANCE).
• What the library prints. For a supplied proxy score array the library's information_report(...).geometric_mean_retention is $\tilde\eta_D=(\det\tilde I_Z/\det\tilde V)^{1/d}$ whenever $\tilde V\succ0$ and rank_rtol projects nothing (O7.1 applies verbatim to any supplied array; the library never knows whether it was given $s$ or $\hat s$). The quantity of interest is $\eta_D$, the true retention of the same labels (O4).
• Not used. $E[s]=0$ is never used in O8.1–O8.3; it is what makes $\eta_D$ the Fisher retention (U1, FI-QUANT-IDENTITY). $E[e]\ne0$ is allowed throughout. Weights are the law. Excluded: the rank projection at singular $\tilde V$ (as in O7); rules refitted on the evaluation sample; $D_s$; sampling variability.

O8.1 Directional reporting budget — [PROJECT_PROVED]

Lemma 1 (conditional Jensen, matrix form). $\mathcal E_Z\preceq\mathcal E$, hence $\varepsilon_Z\le\varepsilon$. Proof. $E[(e-e_Z)(e-e_Z)^\top]\succeq0$ and the cross terms equal $\mathcal E_Z$ by the tower property, so $\mathcal E-\mathcal E_Z\succeq0$. $\square$

Proposition B1. $\tilde I_Z-I_Z=E[c_Ze_Z^\top+e_Zc_Z^\top]+\mathcal E_Z$, and for every $a\in\mathbb R^d$

$$\big|a^\top(\tilde I_Z-I_Z)a-a^\top\mathcal E_Za\big| \le2\sqrt{(a^\top I_Za)(a^\top\mathcal E_Za)},\qquad \big|a^\top(\tilde V-V)a-a^\top\mathcal Ea\big|\le2\sqrt{(a^\top Va)(a^\top\mathcal Ea)} .$$

Proof. Expand $p_b(c_b+e_b)(c_b+e_b)^\top$ and sum; the middle term is $2E[(a\cdot c_Z)(a\cdot e_Z)]$, bounded by Cauchy–Schwarz with $E[(a\cdot c_Z)^2]=a^\top I_Za$ and $E[(a\cdot e_Z)^2]=a^\top\mathcal E_Za$. The second line is the same with $(s,e)$ in place of $(c_Z,e_Z)$. $\square$

Reading. In whitened coordinates a direction $u$ that retains $\rho=u^\top Ru$ is misreported by at most $2\sqrt\rho\,\varepsilon_Z+\varepsilon_Z^2$, the full information by at most $2\varepsilon+\varepsilon^2$. The quadratic error term can add information in any direction; its relative effect is amplified where true retention is small. For a fixed direction with nonzero projected true second moment, equality in Cauchy–Schwarz requires $a^\top e_b=\kappa a^\top c_b$ almost surely; if that second moment vanishes, equality is automatic. This is not vector proportionality.

O8.2 D-retention reporting budget — [PROJECT_PROVED]

Throughout, $U_d(\varepsilon)=2\sqrt d\,\varepsilon+\varepsilon^2$ and, for $\varepsilon<1$, $L_d(\varepsilon)=\max\{2d\log(1-\varepsilon),\ \varepsilon^2-2\sqrt d\,\varepsilon-U_d(\varepsilon)^2/(2(1-\varepsilon)^2)\}$.

Lemma 2 (trace core). For $A=E[cc^\top]\succ0$, $\tilde A=E[(c+e)(c+e)^\top]$, $\mathcal E_A=E[ee^\top]$ and $\varepsilon_A^2=\operatorname{tr}(A^{-1}\mathcal E_A)$: $\operatorname{tr}(A^{-1}(\tilde A-A))=2E[e^\top A^{-1}c]+\varepsilon_A^2$ and $|E[e^\top A^{-1}c]|\le\sqrt d\,\varepsilon_A$, because $E[c^\top A^{-1}c]=\operatorname{tr}(A^{-1}E[cc^\top])=d$. Equality iff $e=\kappa c$ a.s. for a constant $\kappa$.

Lemma 3 (rank-one Loewner bound). For $t\in(0,1)$, $(c+e)(c+e)^\top\succeq(1-t)cc^\top-\frac{1-t}{t}ee^\top$ (the difference is $(\sqrt t\,c+e/\sqrt t)(\sqrt t\,c+e/\sqrt t)^\top$). Hence $\tilde A\succeq(1-t)A-\frac{1-t}{t}\mathcal E_A$, and since $\|A^{-1/2}\mathcal E_AA^{-1/2}\|_{\rm op}\le\varepsilon_A^2$, at $t=\varepsilon_A\in(0,1)$ (zero error is exact equality): $A^{-1/2}\tilde AA^{-1/2}\succeq(1-\varepsilon_A)^2I$.

Lemma 4 (log-det brackets). Let $\lambda_i$ be the eigenvalues of $A^{-1/2}(\tilde A-A)A^{-1/2}$ (all $\ge-1$). Concavity gives $\sum\log(1+\lambda_i)\le\operatorname{tr}(A^{-1}(\tilde A-A))\le U_d(\varepsilon_A)$. For $\varepsilon_A<1$: Lemma 3 gives $\sum\log(1+\lambda_i)\ge2d\log(1-\varepsilon_A)$; and put $m=\min(0,\lambda_{\min})$. The exact integral identity

$$\log(1+\lambda)-\lambda=-\lambda^2\int_0^1\frac{t}{1+t\lambda}\,dt$$

gives $0\ge\log(1+\lambda)-\lambda\ge-\lambda^2/[2(1+m)]$, because $1+t\lambda\ge1+m\ge(1-\varepsilon_A)^2$ throughout the interpolation. The original scalar assertion with positive $\lambda_{\min}$ was false (CE-SCORE-ERROR-LOG-LOWER-001); the final epsilon-based bound is unchanged. Together with $\|A^{-1/2}(\tilde A-A)A^{-1/2}\|_F\le2E[\|c'\|\|e'\|]+\operatorname{tr}E[e'e'^\top]\le U_d(\varepsilon_A)$ (primes: $A^{-1/2}$-whitened) gives the second member of $L_d$. So

$$L_d(\varepsilon_A)\ \le\ \log\det\tilde A-\log\det A\ \le\ U_d(\varepsilon_A), \qquad\text{and}\qquad \log\det\tilde A-\log\det A-\operatorname{tr}(A^{-1}(\tilde A-A))\in\Big[-\tfrac{U_d(\varepsilon_A)^2}{2(1-\varepsilon_A)^2},\,0\Big].$$

Theorem B2 (two-term expansion with curvature remainder). Let $I_Z\succ0$ ($\eta_D>0$), $\tilde V\succ0$, and $\tilde I_Z\succ0$. Then

$$\boxed{\ \log\frac{\tilde\eta_D}{\eta_D}=T_1+T_2+r_3,\qquad T_1=\frac2d\Big(E[e_Z^\top I_Z^{-1}c_Z]-E[e^\top V^{-1}s]\Big),\qquad T_2=\frac{\varepsilon_R^2-\varepsilon^2}{d},\ }$$ $$r_3\in\frac1d\Big[-\frac{U_d(\varepsilon_R)^2}{2(1-\varepsilon_R)^2},\ \frac{U_d(\varepsilon)^2}{2(1-\varepsilon)^2}\Big] \quad(\varepsilon_R,\varepsilon<1),$$

with the bounds

(a) alignment term:

$$|T_1|\ \le\ \frac2d\sqrt{d-\operatorname{tr}R}\;\Big(\varepsilon_Z\sqrt{\tfrac{1-\rho_{\min}}{\rho_{\min}}}+\sqrt{\varepsilon^2-\varepsilon_Z^2}\Big) \qquad |T_1|\ \le\ \frac{2}{\sqrt d}\,(\varepsilon_R+\varepsilon);$$

The two bounds are independent: use their minimum, not a chained ordering (CE-SCORE-ERROR-ALIGNMENT-ORDER-001). The curvature term $r_3$ is second order, not a cubic Taylor remainder; $T_2$ is not the entire quadratic Taylor coefficient. The small-error conditions imply positive definiteness of both proxy matrices.

(b) spurious-information term: $0\le\varepsilon_R^2\le\varepsilon_Z^2/\rho_{\min}$, so $T_2\le(\varepsilon_Z^2/\rho_{\min}-\varepsilon^2)/d$ and $T_2\ge-\varepsilon^2/d$;

(c) crude bracket (no expansion): $\frac1d[L_d(\varepsilon_R)-U_d(\varepsilon)]\le\log(\tilde\eta_D/\eta_D)\le\frac1d[U_d(\varepsilon_R)-L_d(\varepsilon)]$, first order $\pm\frac2{\sqrt d}(\varepsilon_R+\varepsilon)$.

Proof. Apply Lemma 2 and Lemma 4 to $(I_Z,\tilde I_Z,\mathcal E_Z,\varepsilon_R)$ and to $(V,\tilde V,\mathcal E,\varepsilon)$ and subtract; the linear parts are $dT_1$, the $\varepsilon^2$-parts $dT_2$, the curvature parts $dr_3$. For (a) whiten by $V$ ($w=V^{-1/2}s$, $e'=V^{-1/2}e$, $c'_Z=E[w\mid Z]$, $R=E[c'_Zc_Z'^\top]$): $T_1=\frac2d\big(E[e_Z'^\top R^{-1}c'_Z]-E[e'^\top w]\big)$ and $E[e'^\top w]=E[e_Z'^\top c'_Z]+E[(e'-e'_Z)^\top(w-c'_Z)]$ by the tower property, so

$$T_1=\frac2d\Big(E\big[e_Z'^\top(R^{-1}-I)c'_Z\big]-E\big[(e'-e'_Z)^\top(w-c'_Z)\big]\Big).$$

$R^{-1}-I\succeq0$ with operator norm $(1-\rho_{\min})/\rho_{\min}$ and $E[c_Z'^\top(R^{-1}-I)c'_Z]=\operatorname{tr}((R^{-1}-I)R)=d-\operatorname{tr}R$; Cauchy–Schwarz in the $(R^{-1}-I)$ inner product bounds the first expectation by $\varepsilon_Z\sqrt{(1-\rho_{\min})/\rho_{\min}}\cdot\sqrt{d-\operatorname{tr}R}$. For the second, $E\|e'-e'_Z\|^2=\varepsilon^2-\varepsilon_Z^2$ and $E\|w-c'_Z\|^2=\operatorname{tr}(I-R)=d-\operatorname{tr}R$ (Pythagoras). The separate crude inequality of (a) is Lemma 2 applied to each expectation separately. For (b), $I_Z\succeq\rho_{\min}V$ by definition of $\rho_{\min}$, so $I_Z^{-1}\preceq V^{-1}/\rho_{\min}$. (c) is Lemma 4 twice. $\square$

Sharpness of the numerator first-order constant (not the retention-ratio bound). With $e_b=\kappa c_b$ for every cell (the proxy stretches each cell mean by the same factor), Lemma 2 is an equality, $\det\tilde I_Z=(1+\kappa)^{2d}\det I_Z$ and $\varepsilon_R^2=d\kappa^2$, so the numerator gap $2d\log(1+\kappa)$ attains $2\sqrt d\,\varepsilon_R$ to first order, in both signs of $\kappa$. The instrument verifies both identities exactly in $d=1,2,3$ (selftest).

Necessity of the retention scale. No finite uniform relative/log reporting bound from an upper bound on $\varepsilon$ alone exists: CE-SCORE-ERROR-RHO-MIN-NECESSARY-001 is a $d=2$, three-cell, centred atomic family with $\varepsilon^2\in[0.026,0.028]$ uniformly bounded in which $\tilde\eta_D/\eta_D$ runs $1.75\to5.10\to42.8$ as $\rho_{\min}$ runs $0.034\to0.0014\to1.4\cdot10^{-5}$; the weak-direction error scale is $\varepsilon_R^2=4/5,\ 20,\ 2000$. A proxy error of $16\%$ in the whitened metric, aligned with a direction the rule barely retains, reports $75\%$ more retention than there is.

Invertible linear reduction. For fixed labels, $\tilde\eta_D$ is unchanged by $\hat s\mapsto A\hat s$ with $A$ nonsingular. Transform a rule as $q_A(u)=q(A^{-1}u)$ to preserve its labels; feeding transformed scores to an unchanged rule does not do this. Translations are not invariances of uncentred moments (CE-SCORE-ERROR-TRANSLATION-001). Every reporting bound therefore applies with $e_A=A\hat s-s$, with all error scales recomputed for those same labels. The unconstrained least-squares matrix is $A^*=E[s\hat s^\top]\tilde V^{-1}$, and

$$\varepsilon_{\rm lin}^2=d-\operatorname{tr}\big(V^{-1}E[s\hat s^\top]\tilde V^{-1}E[\hat ss^\top]\big) =\sum_{i=1}^d(1-r_i^2),$$

where $r_i$ are uncentred canonical correlations. This identity holds even when $A^*$ is singular; applying the unchanged-retention budget at $A^*$ requires $E[s\hat s^\top]$ nonsingular. Otherwise it only gives the infimum error over invertible maps (by approximation), not an admissible minimizer or a finite bracket at the singular limit (CE-SCORE-ERROR-SINGULAR-LS-001). If $\varepsilon_{\rm lin}<1$, the cross moment is necessarily full rank: one zero canonical correlation would already contribute one unit. The original measured artifact keys eps_aff and table notation denote this linear, zero-intercept reduction.

The deployable direction. The roles of $s$ and $\hat s$ are symmetric ($s=\hat s+(-e)$); with $\tilde\varepsilon^2=\operatorname{tr}(\tilde V^{-1}\mathcal E)$, $\tilde\varepsilon_R^2=\operatorname{tr}(\tilde I_Z^{-1}\mathcal E_Z)\le\tilde\varepsilon_Z^2/\tilde\rho_{\min}$, where $\tilde\rho_{\min}$ is the smallest reported retention eigenvalue (the library's directional diagnostic, INFO-DIRECTIONAL-DIAGNOSTICS), the same expansion bounds $\log(\eta_D/\tilde\eta_D)$. So a user with a reported $\tilde\eta_D$, a reported $\tilde\rho_{\min}$, a reported trace loss, and an error budget in the proxy metric has a bracket for the truth when the stated small-error conditions hold. O8.4 directly bounds the true metric; converting it requires additional spectral control or evaluation of both metrics on a truth-labelled simulation hold-out. Verified exactly in selftest (backward).

Endpoints. At $\eta_D=0$ ($I_Z$ singular, e.g. $K\le d$ at the reference law, U4) the expansion is void and the reported number can be anything in $[0,1]$: the $\rho_{\min}\to0$ limit of the fixture. When $\varepsilon_R\ge1$ the lower bracket is void and the reported information can indeed vanish ($e_b=-c_b$ gives $\tilde I_Z=0$ with $\varepsilon_R^2=d$). Neither is a defect of the bound.

O8.3 Rule transfer under a margin condition — [PROJECT_PROVED]

Now the rule is applied to both scores: $Z=q(s)$, $\hat Z=q(\hat s)$, both retentions computed from the true $s$. $M=\{Z\ne\hat Z\}$, $\pi=P(M)$.

Proposition B3 (i) (mislabel mass). If $q(\hat s)\ne q(s)$ then $\operatorname{dist}(s,\partial C_{q(s)})\le\|e\|$ in any norm — the segment $[s,\hat s]$ is connected, meets $C_{q(s)}$ and its complement, hence its boundary. For a nearest-centre rule with positive-definite metric $G$ and distinct centres, in that same metric the distance from $s$ in cell $b$ to the boundary of $b$ is $\min_{b'\ne b}\big(\|s-\mu_{b'}\|_G^2-\|s-\mu_b\|_G^2\big)/(2\|\mu_b-\mu_{b'}\|_G)$, so its square is rational on rational data and the instrument checks the pointwise statement exactly in the $V^{-1}$ metric. Markov's inequality then gives, with the margin function $M_q(t)=P(\operatorname{dist}_V(s,\partial q)\le t)$,

$$\pi\ \le\ \inf_{t>0}\Big\{M_q(t)+\frac{\varepsilon^2}{t^2}\Big\},\qquad M_q(t)\le Ct^\alpha\ \Rightarrow\ \pi\le\Big(1+\tfrac\alpha2\Big)\Big(\tfrac2\alpha\Big)^{\frac{\alpha}{\alpha+2}}C^{\frac2{\alpha+2}}\,\varepsilon^{\frac{2\alpha}{\alpha+2}}$$

($\alpha=1$: $\pi\le1.89\,C^{2/3}\varepsilon^{2/3}$). Here $C,\alpha>0$, and the minimizing $t_*=(2\varepsilon^2/(\alpha C))^{1/(\alpha+2)}$ must lie in the range of the margin hypothesis; otherwise minimize over the admissible range and cap by one. Zero error gives zero disagreement directly. The powers are correct for $\varepsilon=\|e\|_{L^2(V^{-1})}$; if the input is the squared error, its exponent is $\alpha/(\alpha+2)$. This is the comparison method of Audibert–Tsybakov, §5, with spatial cell-boundary distance replacing posterior distance to the Bayes boundary. Their excess-risk exponent concerns a different, margin-weighted loss.

Proposition B3 (ii) (label-perturbation sandwich). With $c_b$ the $Z$-cell means and $\hat c_b$ the $\hat Z$-cell means of the true score, set either centroid to zero when its own cell has zero mass; use the same finite label alphabet. Then

$$I_{\hat Z}\ \succeq\ I_Z-\Gamma,\qquad I_Z\ \succeq\ I_{\hat Z}-\hat\Gamma,\qquad \Gamma=2E\big[(ss^\top+c_{\hat Z}c_{\hat Z}^\top)\mathbf 1_M\big],\quad \hat\Gamma=2E\big[(ss^\top+\hat c_Z\hat c_Z^\top)\mathbf 1_M\big].$$

Proof. $f=E[s\mid Z]=c_Z$, $f'=E[s\mid\hat Z]=\hat c_{\hat Z}$, within-scatters $W=E[(s-f)(s-f)^\top]$, $W'=E[(s-f')(s-f')^\top]$, and $V=I_Z+W=I_{\hat Z}+W'$. For any $\hat Z$-measurable $g$, $E[(s-g)(s-g)^\top]=W'+E[(f'-g)(f'-g)^\top]\succeq W'$ (Pythagoras: $E[s-f'\mid\hat Z]=0$). Take $g=c_{\hat Z}$, which equals $f$ off $M$: $E[(s-g)(s-g)^\top]=W+E[((s-c_{\hat Z})(s-c_{\hat Z})^\top-(s-c_Z)(s-c_Z)^\top)\mathbf 1_M]\preceq W+E[(s-c_{\hat Z})(s-c_{\hat Z})^\top\mathbf 1_M]\preceq W+\Gamma$. So $W'\preceq W+\Gamma$, i.e. $I_{\hat Z}\succeq I_Z-\Gamma$; swap roles for the other side. $\square$

Corollaries. Trace: $|\operatorname{tr}R_Z-\operatorname{tr}R_{\hat Z}|\le2E[\|w\|^2\mathbf 1_M]+2\pi\max_b\max(\|c'_b\|^2,\|\hat c'_b\|^2)$ (whitened). Determinant: with $\gamma_R=\operatorname{tr}(I_Z^{-1}\Gamma)<1$, $\log\eta_D(\hat Z)-\log\eta_D(Z)\ge\log(1-\gamma_R)$, and symmetrically above; $\gamma_R\le\operatorname{tr}(V^{-1}\Gamma)/\rho_{\min}$ with $\operatorname{tr}(V^{-1}\Gamma)=2E[(\|w\|^2+\|c'_{\hat Z}\|^2)\mathbf 1_M]$. Orders: bounded whitened scores $\|w\|\le B$ give $O(\pi)$ for the trace loss (constant $4B^2$). For a fixed base law and frozen rule with finitely many positive-mass occupied cells, a fourth moment and a uniform bound on the centroids of the compared rules give $O(\sqrt\pi)$ by Cauchy–Schwarz. The centroid condition holds for small perturbations of this fixed partition: original occupied masses remain bounded below; labels originally empty receive only mislabelled points, whose fitted centroid is only used on the reverse side for original points of that label, a null event. Across varying laws or rules, require uniform fourth moments and a common positive lower bound on nonempty base-cell masses. The determinant orders additionally require a uniform retained eigenvalue floor and eventually $\gamma_R<1$. Composing at $\alpha=1$ gives $O(\varepsilon^{2/3})$ for bounded scores and $O(\varepsilon^{1/3})$ with these fourth-moment conditions. The earlier atom-at-$\pi^{-1/2}$ illustration has diverging fourth moment and establishes no sharpness of this fourth-moment rate.

Necessity of the margin. CE-SCORE-ERROR-BOUNDARY-ATOM-001: a scalar centred law with an atom of mass $1/2$ on the boundary of the threshold rule; any proxy lifting it by $\kappa>0$ has $\pi=1/2$, $\eta_D$ jumps $4/5\to1/2$, and $M_q(t)\ge1/2$ for all $t>0$ makes (i) vacuous. The sandwich holds with $\Gamma=4$ — the budget is a function of the mislabel mass, and nothing makes the mass small.

O8.4 Classifier calibration to score error (OPEN-CLASSIFIER-CALIBRATION-FI) — [BRIDGE]

Per O3 (CLASSIFIER-MIXTURE-SCORE-FORMULA), with posteriors $\eta\in\Delta^{m}$ under training priors $\pi$, $r_\alpha=\eta_\alpha/\pi_\alpha$, $D(\eta)=\sum_\beta\theta_{0\beta}r_\beta$ and $\Phi_\alpha(\eta)=r_\alpha/D(\eta)$.

Proposition B4. For $\theta_0$ interior, on the simplex: $D(\eta)\ge D_{\min}:=\min_\beta\theta_{0\beta}/\pi_\beta>0$ (since $\sum\eta_\beta=1$), $0\le\Phi_\alpha\le1/\theta_{0\alpha}$ (since $D\ge\theta_{0\alpha}r_\alpha$), and for $\eta,\eta'\in\Delta^m$

$$|\Phi_\alpha(\eta')-\Phi_\alpha(\eta)|\ \le\ \frac1{D_{\min}}\Big(\frac{|\eta'_\alpha-\eta_\alpha|}{\pi_\alpha}+\frac1{\theta_{0\alpha}}\sum_\beta\frac{\theta_{0\beta}}{\pi_\beta}|\eta'_\beta-\eta_\beta|\Big),$$

from $\Phi'-\Phi=[(r'-r)-\Phi(\eta)(D'-D)]/D'$. Hence $\|\Phi(\eta')-\Phi(\eta)\|_2\le L\|\eta'-\eta\|_2$ with $L=\frac1{D_{\min}}\big(\frac1{\pi_{\min}}+Q\sqrt m\,\|1/\theta_0\|_2\big)$, $Q=\max_\beta\theta_{0\beta}/\pi_\beta$. The score is a fixed linear image $s=T\Phi$ (for a normalized mixture, use the tangent chart, e.g. component-minus-reference differences; $T\mathbf1=0$), so with $\hat s=T\Phi(\hat\eta)$:

$$\varepsilon^2\ \le\ \|V^{-1}\|_{\rm op}\,\|T\|_{\rm op}^2\,L^2\;E_{P_{\theta_0}}\|\hat\eta-\eta\|^2 \ \le\ \|V^{-1}\|_{\rm op}\,\|T\|_{\rm op}^2\,L^2\,Q\;E_{P_\pi}\|\hat\eta-\eta\|^2,$$

the second step because $dP_{\theta_0}/dP_\pi=\sum\theta_{0\beta}\phi_\beta/\sum\pi_\beta\phi_\beta\le Q$. Under the training mixture $P_\pi$ with one-hot component label $Y$, $E\|\hat\eta-\eta\|^2=\mathrm{BS}(\hat\eta)-\mathrm{BS}(\eta)$ is the excess Brier score ($\eta=E[Y\mid X]$), and it splits orthogonally as

$$E\|\hat\eta-\eta\|^2=\underbrace{E\|\hat\eta-E[\eta\mid\hat\eta]\|^2}_{\text{reliability (calibration error)}} +\underbrace{E\|\eta-E[\eta\mid\hat\eta]\|^2}_{\text{resolution gap}} .$$

Reliability is a population functional that can be estimated from independent labelled data using a justified calibration estimator; naive finite binning/isotonic fits are not an exact certificate. Excess Brier error requires the Bayes reference score or truth posteriors, which simulation can supply.

For completeness, the inverse map on the image is $\eta_a=\pi_a\Phi_a/h$, $h=\sum_b\pi_b\Phi_b\in[1/Q,1/D_{\min}]$. Subtracting two ratios gives the explicit bound

$$\|\eta'-\eta\|_2\le L_{\rm inv}\|\Phi'-\Phi\|_2,\qquad L_{\rm inv}=Q(\pi_{\max}+\|\pi\|_2),$$

using $\|\eta\|_2\le1$. Thus posterior and full component-ratio errors are equivalent. They are not equivalent to score error for an arbitrary chart. Write $H=\{h:\theta_0^\top h=0\}$ and $\sigma_T=\inf_{h\in H,\|h\|_2=1}\|Th\|_2$. Only if $\sigma_T>0$ and $V\succ0$,

$$\varepsilon^2\ge \frac{D_{\min}\sigma_T^2}{L_{\rm inv}^2\|V\|_{\rm op}} E_{P_\pi}\|\hat\eta-\eta\|^2 \ge\frac{D_{\min}\sigma_T^2}{L_{\rm inv}^2\|V\|_{\rm op}}\,\mathrm{reliability}.$$

The full free-fraction chart is injective on $H$; reduced parameter submodels need not be (CE-CLASSIFIER-CALIBRATION-CHART-001). This lower bound includes constants and concerns squared error, not retention distortion.

Consequence. Excess Brier error gives an upper score-error budget, which O8.2 converts into a conditional reporting bound. Calibration alone does not certify small error because the resolution gap is unobserved. Conversely, bad calibration does not certify distortion of reported retention: even a full chart can produce an invertible rescaling of the true score (CE-CLASSIFIER-CALIBRATION-RETENTION-001). The forward constants remain valid. The original selftest tested 300 simplex pairs / 940 coordinates, not 940 pairs, and did not test either false lower-bound inference. This is a bridge from the published Brier decomposition, with explicitly checked project constants.

O8.5 AUC is not enough — [COUNTEREXAMPLE]

CE-AUC-INVARIANT-PROXY-RETENTION-001: the scalar law $s\in\{-2,-1,1,2\}$, equal weights, threshold rule at $0$, true retention $9/10$. Two strictly increasing distortions of the score — the same ranking, the same ROC curve for every threshold, the same family of rank-cut labels under corresponding thresholds (and the same fixed zero-threshold labels), the same true retention — report $100/101$ (compress within cells) and $5105/10006$ (stretch one atom). In $d=1$ a strictly monotone $g$ preserves rank cuts after the thresholds are transformed; it need not preserve a fixed numerical threshold. It moves the reported uncentred second-moment ratio of $g(s)$ across an interval; with unequal atom weights arbitrarily close to $0$. Ranking quality is not the relevant classifier metric; the whitened $L^2$ error is (O8.2), and it is not a function of the ROC curve.

O8.6 Deployment corollary — [PROJECT_PROVED]

Let $\tilde\eta_D(q)$ be the reported retention of the labels $q(\hat s)$, $\eta_D(q\circ\hat s)$ their true retention and $\eta_D(q\circ s)$ the true retention of $q(s)$. For the rule $\hat q$ fitted on the proxy and any comparison rule $q$ (e.g. the oracle optimum on $s$) with $\tilde\eta_D(\hat q)\ge\tilde\eta_D(q)$ — true of the global proxy optimum, and of an exchange-stable solution only if it happens to beat $q$ on the proxy objective —

$$\log\eta_D(\hat q\circ\hat s)\ \ge\ \log\eta_D(q\circ s)\;-\;\beta^+(\hat q)\;-\;\beta^-(q)\;+\;\log\big(1-\gamma_R(q)\big),$$

where, for $g(q)=\log(\tilde\eta_D(q)/\eta_D(q\circ\hat s))$, define $-\beta^-(q)\le g(q)\le\beta^+(q)$ using O8.2, and assume all retentions in this chain are positive and the transfer budget $\gamma_R(q)<1$. The correct chain is

$$\log\eta_D(\hat q\circ\hat s) \ge\log\tilde\eta_D(\hat q)-\beta^+(\hat q) \ge\log\tilde\eta_D(q)-\beta^+(\hat q) \ge\log\eta_D(q\circ\hat s)-\beta^-(q)-\beta^+(\hat q).$$

Apply O8.3 to finish. The over-reporting budget belongs to the selected rule; the under-reporting budget belongs to the comparator. There is no free substitution $q(s)\mapsto q(As)$ for a fixed oracle comparator. A change of score coordinates must transport the rule, the metric, and the comparator together; otherwise it is a new comparison with a new target. The corollary still needs truth-based budgets for two rules and proxy-objective dominance; exchange stability alone supplies neither.

O8.7 Measured (protocol D, run before and after the derivation) — [MEASURED]

Instrument py/score_error_budget.py; records in artifacts/SCORE-ERROR-BUDGET/.

• selftest (exact, 400 random atomic laws, $d\in\{1,2,3\}$, $K\in\{2,3,4\}$, $N\le8$, unequal weights, duplicate atoms, singleton cells, constant error shifts, three error scales; 22 942 checks; $0$ failures): Lemma 1, B1 on axis and random directions, the Lemma 2 cores for numerator and denominator, Lemma 3 at $t\in\{1/4,1/2,3/4\}$, the float brackets of Lemma 4 and of B2(c), the exact $T_1$ against each B2(a) bound separately and the remainder bracket, $\varepsilon_R^2\le\varepsilon_Z^2/\rho_{\min}$, the linear identity and admissible invariance, the backward direction, B3 (i) pointwise on nearest-centre rules in the $V^{-1}$ metric, B3 (ii) both sandwiches and the trace corollary on random second labellings, B4 on 300 simplex pairs (940 coordinate checks); the sharpness attainer identities in $d=1,2,3$ ($\kappa=\pm1/5$).

• door3 (the classifier example; examples/door3_classifier.py rungs rebuilt through the O6 audit's closed forms; all population quantities by cut-aligned QUADPACK; $d=1$, $K=4$):

  rung	$\eta_D$ true	reported	$\log$ ratio	$\varepsilon$ / $\varepsilon_{\rm aff}$	$\varepsilon_R$ / aff	$T_1$	B2(a) sharper / aff	B2 crude / aff	$1-\eta_D$	$\pi$	B3 (i) bound
  15	0.8937	0.9671	0.0789	0.500 / 0.416	0.447 / 0.347	0.084	0.270 / 0.240	1.89 / 1.53	0.106	0.275	1.00
  60	0.9557	0.9710	0.0159	0.289 / 0.235	0.262 / 0.200	0.025	0.079 / 0.073	1.10 / 0.87	0.044	0.186	1.00
  300	0.9716	0.9738	0.0023	0.096 / 0.067	0.091 / 0.061	0.0028	0.016 / 0.013	0.37 / 0.26	0.028	0.051	0.45

  Reading: the door3 classifier's whitened score error is large (50% at the smallest rung) while the reported retention is only 8% off — the two cancellations of B2(a) at work (a mostly linear error, uncentred correlation $0.909\to0.998$; a nearly lossless rule, $\sqrt{1-\eta_D}=0.33$). The crude first-order budget over-covers by $24\times$, the sharper alignment bound by $3$–$6\times$, and $T_1$ itself tracks the gap to within the curvature remainder. The rule applied to the true score retains more ($0.947,0.971,0.973$) than the proxy labels ($0.894,0.956,0.972$); both sandwiches hold ($\Gamma=0.80,0.41,0.10$). The margin bound (i) is vacuous at rungs 15 and 60 ($\varepsilon\ge0.29$) and $0.45$ against $\pi=0.051$ at rung 300: B3 (i) is informative only for $\varepsilon\lesssim0.1$. Recorded values of O6/O7 for $\eta_D$ and the proxy reproduce to rounding.

• synthetic2d (the O7 bounded three-component mixture, $d=2$, $K=4$, $\eta_D=0.3303$, $\rho=(0.131,0.830)$; proxies: a mis-specified template with all component means shifted by $\delta\in\{0.05,0.1,0.2,0.4\}$, and at each matched $\varepsilon\in\{0.072,0.143,0.285,0.563\}$ an error of that size aligned with the weakest and with the strongest retained whitened direction, sign-correlated with the cells): every row inside the B2(c) bracket; $\log(\tilde\eta_D/\eta_D)$ for the weak-direction proxy $+0.021,+0.068,+0.204,+0.468$ against $+0.004,+0.009,+0.022,+0.052$ for the strong-direction proxy at the same $\varepsilon$ — the gap here is the $T_2$ term ($\varepsilon_R=0.20,0.39,0.78,1.54$ versus $0.08,0.17,0.33,0.66$), $T_1$ stays below $0.04$ on every row, and $(\varepsilon_R^2-\varepsilon^2)/2$ reproduces the weak-direction gap to $0.01$ at $\varepsilon\le0.29$. The template shift admits a linear least-squares reduction ($\varepsilon_{\rm aff}\approx\varepsilon$, canonical correlations $\ge0.79$) and reports within $\pm0.06$ up to $\varepsilon=0.29$.

O8.8 Self-adversarial pass (protocol G) and verdict (protocol H)

• Ties, duplicates, singleton cells: all statements are about laws; the instrument's grid includes duplicate atoms and singleton cells. Empty cells contribute nothing ($0/0:=0$) and are excluded from every sum. Singular information: B1 and B3 need nothing beyond finite second moments; a finite B2 logarithm needs $I_Z,\tilde I_Z,\tilde V\succ0$ (endpoints above); the library's rank projection is outside every statement, as in O7. Nuisance/$D_s$: not covered. Atomic laws: the proofs are for arbitrary laws; every fixture is atomic. Hidden compactness: none in B1, B2, B3 (ii); B3 (i)'s exponent needs a margin function, and bounded or fourth-moment scores set the $\pi$-to-retention order. First-order to finite: every bracket is non-asymptotic; the "first order" readings are the $\varepsilon\to0$ forms. Empirical to population: the inequalities hold for every finite law, hence sample-wise for the library's number when both scores are on the sample; no limit is taken. $E[e]\ne0$: allowed and exercised. Weights: the law. Score estimation error: the subject. New events: the rule is frozen; O8.3 is the new-event statement. Dependence of $q$ on $\hat s$: irrelevant to O8.1–O8.2 (labels are labels), handled in O8.6 by the proxy-optimality step, which is the one place where an exchange-stable (non-global) solution weakens the statement.
• Information-loss implication (H). O8.2 bounds $\log(\tilde\eta_D/\eta_D)$ — reported versus true geometric-mean retention of the deployed labels — by the alignment term $\propto\sqrt{d-\operatorname{tr}R}$ and the spurious-information term $\propto\varepsilon_Z^2/\rho_{\min}$; O8.1 does it per direction; O8.3 bounds the loss of applying the rule to the proxy instead of the truth; O8.6 composes them into the deployed-versus-oracle loss. Train-only versus held-out is not the axis here (O7 is).
• Verdict. Proved (B1, B2, B3, B6 project level; B4 bridge; B5 fixture). OPEN-SCORE-PERTURBATION is settled at first order for a frozen rule; OPEN-CLASSIFIER-CALIBRATION-FI is half-settled: the chain from posterior error to score error is one inequality, and a calibration-to-score lower bound needs an injective chart and explicit constants; it is not a lower bound on retention distortion. Open remainder, backlog not theme: a uniform-over-rules budget together with a proxy optimization-gap certificate (uniform perturbation control alone does not make an exchange-stable solution globally optimal); the $D_s$/profiled retention; refitted rules; second-order sharp constants for $r_3$; the resolution gap without truth scores (OPEN-REPRESENTATION-LOSS-ESTIMATION's cross-fitting).
• Product consequence. Closure step 1's docs sentence must say: the printed retention for a classifier provider is the reported number; its distance to the true retention is at first order $\frac2d\sqrt{d-\operatorname{tr}R}\,(\dots)$ plus $\varepsilon_Z^2/(d\rho_{\min})$ in the classifier's whitened score error, small for a nearly lossless rule with a well-calibrated, well-resolved classifier, and not controlled by AUC; the O7 error bar covers sampling only. OPEN-PARAMETER-MISMATCH (closure step 4): a score shift $e=s_{\theta_0+\delta}-s_{\theta_0}$ under the reference law is an instance of O8.2; the law shift is not, and remains step 4's content.