Global profiled optima can lose conditioning

Audit (30 Aug 2026, AUDITS/AUDIT-DS-MARGINS-AT-OPTIMA-001.md): verified with hardened assumptions. The originally registered generality — arbitrary $d_\lambda$ under the bare cardinality assumption $K\ge3$ — is false: for $K=d_\lambda+1$ exact centering forces $\operatorname{rank}(I_z)\le K-1$, so every feasible labeling has profiled value exactly $0$ while $v_K>0$ (COUNTEREXAMPLES/CE-DS-MARGINS-RANK-VACUITY-001.json). The theorem below is therefore stated for $d_\lambda=1$, where $K\ge3$ is exactly $K\ge d_\lambda+2$; the $d_\lambda\ge2$, $K\ge d_\lambda+2$ branch is open (Proposition 6's steering is one-dimensional in the nuisance). The audit supplied the previously missing pieces of Propositions 5–6 and replaced the misattributed Glivenko–Cantelli import in the proof of conclusion (3); see the audit report §8.

Setting: $S=(S_\psi,S_\lambda)\sim P$ on $\mathbb R^{2}$ ($d_\psi=d_\lambda=1$; general $d_\psi$ and general $d_\lambda$ are discussed at the end), $E S=0$, $E\|S\|^2<\infty$, full information $I=E[SS^\top]\succ0$. Efficient score $\hat s=S_\psi-B^*S_\lambda$, $B^*=I_{\psi\lambda}I_{\lambda\lambda}^{-1}$, $\sigma_s^2=E[\hat s^2]=S_\psi(I)$. All second moments are uncentered about the origin (empirical scores exactly centered, as everywhere in this chapter).

Law class.

• (L) conditional centering: $E[S_\lambda\mid\hat s]=0$ a.s. Since $\operatorname{Cov}(\hat s,S_\lambda)=0$ always, (L) upgrades uncorrelatedness to mean-independence. It holds for jointly Gaussian scores, for every elliptical law with finite second moments (conditional means are linear), and more generally whenever the regression of the nuisance score on the efficient score vanishes.
• (S) scalar regularity: $\operatorname{law}(\hat s)$ is atomless with a positive density near the optimal cell boundaries, and the optimal $K$-point quantizer of $\operatorname{law}(\hat s)$ under squared error is unique (strict log-concavity suffices — Gaussian in particular; see the an earlier, since-closed question literature audit for the uniqueness citations).
• (R) swap richness: for some $0<\ell<L$, $P(S_\lambda\in[\ell,L]\mid \hat s)$ and $P(S_\lambda\in[-L,-\ell]\mid\hat s)$ are bounded below near the optimal boundaries (automatic for Gaussian, where $S_\lambda\perp\hat s$).

Scalar quantities: $W_K$ = optimal $K$-cell SSE of $\operatorname{law}(\hat s)$, $v_K=\sigma_s^2-W_K$, $J^*$ its optimal interval partition, with masses $w_b^*>0$ and distinct centroids $c_b^*$.

For a quantizer $q$ (or finite labeling $z$): $W_b$, moments $m_b=E[S1_{q=b}]$, $I_q=\sum_b m_bm_b^\top/W_b$, profiled value $\Phi(q)=S_\psi^+(I_q)$ in the DS11 pseudo-inverse extension ($=$ the in-bin Schur value whenever $I_{q,\lambda\lambda}\succ0$).

Theorem (dichotomy)

Let $z^{(N)}$ be exact global finite $D_s$ optima of i.i.d. samples from $P$ satisfying (L)+(S)+(R), over feasible $K$-cell labelings ($K\ge3=d_\lambda+2$ — load-bearing: at $K=2=d_\lambda+1$ the rank boundary above makes every feasible value exactly $0$; equal weights; nonsingular binned nuisance block). Then, almost surely:

1. (Value; unrestricted supremum.) $\hat\Phi_s(z^{(N)})\to v_K=\sup_q S_\psi^+(I_q)$, the supremum over all measurable $K$-cell quantizers. The supremum is attained exactly at the optimal $\hat s$-interval quantizer $J^*$ — and at nothing else — and $J^*$ is fully nuisance-degenerate: $m_\lambda(J^*_b)=0$ for every $b$, so $I_{J^*}$ has zero nuisance row and block, $\lambda_{\min}(I_{J^*})=0$, and $J^*$ is DS9-infeasible (the binned model carries no nuisance information).
2. ((M2) holds.) $\min_b\hat W_b(z^{(N)})\to\min_b w_b^*>0$: the cell-mass margin is automatic asymptotically; singleton cells die out. The $N\le18$ singleton evidence is pre-asymptotic.
3. ((M3) fails.) $\hat I_{\lambda\lambda}(z^{(N)})\to0$, $\hat I_{\psi\lambda}(z^{(N)})\to0$, hence $\lambda_{\min}(\hat I_N(z^{(N)}))\to0$: for every $\kappa>0$ the conditioning margin fails eventually, for every law in the class. The optimizer sheds binned nuisance information by design.
4. (Margin-compatible optimum is strictly suboptimal.) For every $\kappa>0$, $v^*(\kappa):=\sup\{\Phi(q):\lambda_{\min}(I_q)\ge\kappa\} <v_K$. DS14's margin hypothesis set is empty along free global-optimum sequences: the conditional bridge governs margin-certified (necessarily $\delta(\kappa)$-suboptimal) solutions, never free global optima, on this law class.
5. (Domination equality at optima.) The DS11(a) gap at $z^{(N)}$ — $(\hat B^*_N-\hat B_{z})\hat I^z_{\lambda\lambda}(\cdot)^\top$ — tends to $0$: efficient-score domination becomes an equality along global optima (answering the fourth an earlier, since-closed question sub-question).

Proof

Lemma 1 (scalar reduction; any $P$ with finite second moments). For every measurable $K$-cell $q$: $\Phi(q)\le\sum_bW_bE[\hat s\mid b]^2\le v_K$. Proof. The DS11 variational form gives $\Phi(q)=\min_B\sum_bW_b(\mu_{\psi,b}-B\mu_{\lambda,b})^2 \le\sum_bW_bE[\hat s\mid b]^2$ at $B=B^*$. For the second inequality, $\sum_bW_bE[\hat s\mid b]^2=E\hat s^2-\sum_bE[(\hat s-E[\hat s\mid b])^21_b]$, and reassigning every point to the nearest of the $K$ conditional means decreases the SSE term while producing an $\hat s$-measurable partition, so the SSE is at least $W_K$. ∎

Lemma 2 (attainment iff degenerate; (L)). Under (L), every $\hat s$-measurable partition has $m_{\lambda,b}=E[E[S_\lambda\mid\hat s]1_{\hat s\in J_b}]=0$; its variational form is $B$-independent and equals $\sum_bW_bE[\hat s\mid b]^2$ (as $\mu_{\psi,b}=E[\hat s\mid b]$). At $J^*$ this is $v_K$, matching Lemma 1's bound: $\sup_q\Phi=v_K$. Equality analysis: equality in Lemma 1's SSE step forces the cells to be nearest-centroid in $\hat s$ up to null sets with an optimal centroid set; under (S) the optimum is unique, so any attainer is $J^*$ a.e. ∎

Lemma 3 (rigidity). Under (L)+(S), for every $\varepsilon>0$ there is $\delta>0$: any $K$-cell $q$ with $\sum_bW_bE[\hat s\mid b]^2\ge v_K-\delta$ has (after relabeling) $\max_bP(A_b\,\Delta\,\{\hat s\in J^*_b\})\le\varepsilon$, and all masses $\ge\eta(\delta)>0$. Consequently $|m_{\lambda,b}|\le \|S_\lambda\|_2\sqrt\varepsilon$ (Cauchy–Schwarz over the symmetric difference, using $E[S_\lambda1_{J^*_b}]=0$) and $\|I_{q,\lambda\lambda}\|\le C\varepsilon$. In particular $v^*(\kappa)<v_K$ for every $\kappa>0$. Proof. (i) Effective cells. For $\eta>0$ call $b$ active if $W_b\ge\eta$; inactive cells contribute at most $\sup_{P(B)\le K\eta}E[\hat s^21_B]=:\tau(K\eta)\to0$ to the between-value (Cauchy–Schwarz then uniform integrability). (ii) Deletion comparison. Assigning inactive mass to the active centroid closest to the origin (bounded by $\sigma_s\sqrt{K/(1-K\eta)}$) shows the active centroid set $C_a$ satisfies $E[\min_{c\in C_a}(\hat s-c)^2]\le W_K+\delta+\rho(\eta)$ with $\rho(\eta)\to0$. (iii) Full support. Strict monotonicity $W_{K-1}>W_K$ (atomless law, infinite support) forces all $K$ cells active once $\delta+\rho(\eta)<W_{K-1}-W_K$: the mass bound. (iv) Centroid convergence. Compactify: a centroid escaping to infinity has vanishing Voronoi mass, contradicting (iii)'s budget via $W_{K-1}>W_K$; so along any sequence of $\delta_m$-near-optimal partitions the centroid sets converge to the unique optimal $C^*$ (continuity + uniqueness). (v) Partition convergence. The excess of $q$ over its own centroids' nearest-point rule bounds the misassigned mass at distance $\ge t$ from the $C^*$-midpoints; boundary $t$-slabs have small mass (atomlessness); diagonalize. ∎

Proposition 4 (exact empirical sandwich; finite algebra). For every sample and every feasible labeling $z$: with $\hat s_N$ built from the full-sample empirical regression $\hat B^*_N$ (normal equations: $\sum_iw\hat s_{N,i}s_{\lambda,i}=0$),

$$\hat\Phi_s(z)\;=\;\mathrm{btw}(\hat s_N;z)- \hat c(z)^\top\hat I^{z\,-1}_{\lambda\lambda}\hat c(z) \;\le\;\mathrm{btw}(\hat s_N;z)\;\le\;\hat v_K,$$

where $\mathrm{btw}(x;z)=\sum_b(\sum_{i\in b}wx_i)^2/\hat W_b$, $\hat c(z)=$ the binned $(\hat s,\lambda)$ cross-moment vector, and $\hat v_K$ is the exact optimal $K$-grouping value of the $\hat s_N$ sample — attained by intervals of sorted $\hat s_N$ (1-D contiguity), hence computable exactly by enumeration/DP. Both inequalities are verified in exact rational arithmetic for every optimum in the N-DS-MARGINS-TREND suite.

Proposition 5 (bracket limits). A.s. $\hat v_K(\hat s_N)\to v_K$: value convergence of empirical scalar quantization (uniform SLLN over interval classes; Pollard 1981 — value convergence needs no uniqueness), plus a uniform-in-labelings Lipschitz bound in the tilt to absorb $\hat B^*_N\to B^*$ and the exact centering. Proof of the Lipschitz bound (audit-supplied). For any labeling $z$, $\mathrm{btw}(s_\psi-\beta s_\lambda;z)=A_z-2\beta B_z+\beta^2C_z$ with $A_z,B_z,C_z$ the binned between-moments; $0\le C_z\le\sum_iw s_{\lambda,i}^2$ and $|B_z|\le\sqrt{A_zC_z}\le\sqrt{\sum_iws_{\psi,i}^2\sum_iws_{\lambda,i}^2}$ (between-values are dominated by total second moments; per-cell Cauchy–Schwarz for the cross term). Hence on $|\beta-B^*|\le1$ every $z\mapsto\mathrm{btw}$ is $\hat L$-Lipschitz in $\beta$ with one SLLN-bounded $\hat L$, so their maximum $\hat v_K(\beta)$ is too, and $|\hat v_K(\hat B^*_N)-\hat v_K(B^*)|\le\hat L|\hat B^*_N-B^*|\to0$. The exact centering shifts every between-value by the same $-\bar{\hat s}_N^2\to0$. ∎

Proposition 6 (achievability by steering). Under (L)+(S)+(R), a.s. there are feasible labelings $z'_N$ with $\hat\Phi_s(z'_N)\ge\hat v_K-O(N^{-3/4}\sqrt{\log\log N})$. Proof (the tax obstruction and its resolution; the audit hardened the original sketch — the four supplied ingredients follow the construction). Write $x_b=m_{\hat s,b}/\hat W_b^{1/2}$, $y_b=m_{\lambda,b}/\hat W_b^{1/2}$ ($d_\lambda=1$); Proposition 4 reads $\hat\Phi_s(z)=|x|^2-\langle x,y\rangle^2/|y|^2$: the profiled value pays the squared projection of $x$ onto the direction of $y$. At the DP-optimal interval labeling $z_0$ of $\hat s_N$, both $\langle x,y\rangle$ ($=-$within-cell cross-moment, mean zero under (L)) and $|y|$ fluctuate at the $N^{-1/2}$ scale, so the tax is a $\Theta_p(1)$ random ratio — the plain interval labeling does not prove achievability, and one-dimensional cancellations (tilting the slope; greedily shrinking $|\langle x,y\rangle|$) leave the ratio $\Theta_p(1)$. The resolution is to steer the vector $y$ itself: single-point swaps between $\hat s$-adjacent cells, using points inside boundary slabs of width $N^{-1/4}$ with $\lambda$-magnitude in $[\ell,L]$ and chosen sign (available in ample number by (R) and the LLN), move $y$ in two independent directions of its constraint plane $\{\sum_b\hat W_b^{1/2}y_b=0\}$ with increments $\Theta([\ell,L]/N)$ and individual between-cost $\le2\Delta t/N+O(N^{-2})$. Steer $y$ to within $O(L/N)$ of a target $y^*$ with $\langle x,y^*\rangle=0$ and $|y^*|=N^{-1/2}$ (the intersection of the constraint plane with $x^\perp$ is nonempty for $K\ge3$, and $x\ne0$ since $\mathrm{btw}(z_0)=\hat v_K>0$): then $|\langle x,y\rangle|=O(1/N)$ while $\hat I^z_{\lambda\lambda}=|y|^2=N^{-1}(1+o(1))$ is constructed, not random — no small-ball event needed — so the tax is $\tilde O(1/N)$, the between-value loss from $\tilde O(N^{1/2})$ swaps is $\tilde O(N^{-3/4})$, and feasibility $\hat I_{\lambda\lambda}>0$ holds by construction.

The four ingredients the audit supplied to close the sketch:

• (Boundary and mass consistency.) The empirical DP boundaries and masses of $z_0$ converge a.s. to those of $J^*$ — from (S)-uniqueness and Pollard's argmin consistency for the empirical scalar problem; this both places the swap slabs at the population boundaries where (R) applies and keeps the $\sqrt{\hat W_b}$ scalings bounded, so the two adjacent-swap increment directions $v_{b,b+1}=e_{b+1}/\sqrt{\hat W_{b+1}} -e_b/\sqrt{\hat W_b}$ stay uniformly non-collinear (greedy two-direction steering then reaches any plane target to within one increment).
• (Availability count.) The number of sample points in a data-dependent slab of width $2N^{-1/4}$ around an empirical boundary carrying each nuisance-sign window is $\ge cN^{3/4}$ eventually a.s.: mean $\asymp N^{3/4}$ by (S) positive density and (R), fluctuation $O(\sqrt{N\log\log N})$ uniformly over the fixed VC class of intervals $\times$ sign windows, and the data-dependent tilt $\hat B^*_N$ shifts memberships by $L|\hat B^*_N-B^*| =O(\sqrt{\log\log N/N})\ll N^{-1/4}$ (LIL). The pools ($\asymp N^{3/4}$) dominate the $\tilde O(N^{1/2})$ swaps, each point used once, sources never emptied (masses stay near $w^*>0$).
• (Drift accounting.) Each swap moves $x$, the centroids, and the midpoints by $O(1/N)$; over $\tilde O(N^{1/2})$ swaps the cumulative $\tilde O(N^{-1/2})$ drift changes $\langle x,y\rangle$ by $O(|\Delta x||y|)=\tilde O(N^{-1})$ and each swap's between-cost bound by a factor $1+o(1)$ — absorbed.
• (Steering distance.) $|y(z_0)-y^*|=O(\sqrt{\log\log N/N})$ a.s. (LIL over the fixed interval class under (L)), whence the $\tilde O(N^{1/2})$ swap count and the stated $\tilde O(N^{-3/4})$ rate; the theorem consumes only $o(1)$. ∎

Proof of the theorem. (1) Upper: Propositions 4–5. Lower: global optimality against Proposition 6. Attainment/uniqueness: Lemma 2. (2)–(3): the sandwich squeezes $\mathrm{btw}(\hat s_N;z^{(N)})\to v_K$; Lemma 3 applied along the empirical scalar problem (Pollard's argmin-continuity carries the uniqueness/compactness argument to $\hat\mu_N(\hat s_N)\Rightarrow\operatorname{law}(\hat s)$) forces the optimum's cells to converge in sample measure to $J^*$: masses converge to $w^*$ (conclusion 2), and the empirical cell nuisance moments converge to $E[S_\lambda1_{J^*_b}]=0$ (empirical Cauchy–Schwarz over the symmetric differences plus the LLN on the fixed sets $J^*_b$). The data-dependent slope is absorbed by a Glivenko–Cantelli law over the fixed VC class of half-planes $\{s:s_\psi-\beta s_\lambda\le c\}$, $(\beta,c)$ in a compact neighborhood of $(B^*,\cdot)$, with the population side needing only (S)-atomlessness of $\operatorname{law}(\hat s)$ at the $K-1$ fixed optimal boundaries — not the (M4) slab margin or the (M5) scale that power the superficially similar step in audit §8 of AUDITS/AUDIT-DS-POPULATION-BRIDGE-001.md (the earlier citation of that step here was a misattribution; the audit report §8 records the corrected lemma): $\hat I_{\lambda\lambda},\hat I_{\psi\lambda}\to0$, and $\lambda_{\min}\le\hat I_{\lambda\lambda}$ (conclusion 3). (4): if $\Phi(q_m)\to v_K$ with $\lambda_{\min}(I_{q_m})\ge\kappa$, Lemma 1 gives $\mathrm{btw}\to v_K$, Lemma 3 gives $I_{q_m,\lambda\lambda}\to0<\kappa$ — contradiction. (5): the gap equals the tax $\hat c^\top\hat I_{\lambda\lambda}^{-1}\hat c= \mathrm{btw}-\hat\Phi_s\to v_K-v_K=0$. ∎

Interpretation: the in-bin optimum escapes to the projected formulation

The theorem says the free $D_s$ optimizer sheds the in-bin formulation's own feasibility margin: its limit $J^*$ is exactly the optimal binning of the projected full-data efficient score (the external-projection problem, DS-PROJECTED-K-REQUIREMENT), which is infeasible as an in-bin profiled model in the sense of DS9's feasibility split — its binned nuisance block is exactly singular, so the in-bin formulation cannot estimate the nuisance from the bins. (Audit note: DS9's recorded statement is the $K\le d$ rank ceiling; $J^*$'s infeasibility is the zero-nuisance-block face of the same split at any $K$, not the cardinality face.) The two formulations, kept deliberately separate by invariant 3, merge at the optimum — and that merger is why the margins fail. This is the asymptotic form of the tie fixture's finite phenomenon (pseudo-inverse value $1191/4096$ above the feasible optimum $1083/4096$), and the partition-side analogue of the design-theory fact that $D_s$-optimal designs can carry singular information matrices (Silvey 1978).

Deployability

For conditionally centered (in particular Gaussian/elliptical — the linear template fit) score laws with $d_\psi=1$:

• The correct compile target for the profiled criterion is the scalar efficient-score interval rule: estimate $\hat B^*_N$ from the full sample, bin $\hat s$ by the exact DP. Certify (a) slope stability and (b) the scalar quantizer's own 1-D margins (mass, boundary separation) — all easy — instead of the DS14 (M3) certificate, which this theorem proves can never hold at free global optima.
• A margin-certified in-bin quantizer (audit §11 flow) is legitimate but costs at least $\delta(\kappa)=v_K-v^*(\kappa)>0$ of profiled information — the certificate has a quantified price (information-loss implication: $\eta_{D_s}$-retention strictly below the unrestricted optimum).
• The binned model at the compile target deliberately carries no nuisance information (DS9): the report must state that nuisance estimation stays unbinned/full-sample — matching the CMS-2025 use pattern (profile at the full-likelihood level, bin for the POI).

What is deliberately not claimed

(i) Nothing beyond class (L): if $E[S_\lambda\mid\hat s]\ne0$, the tax has a $\Theta(1)$ population component on $\hat s$-intervals and the supremum may be attained at nondegenerate quantizers — the margins may then hold; this is OPEN-DS-MARGINS-NONCENTERED (OPEN-DS-MARGINS-NONCENTERED). (ii) $d_\psi>1$: Lemma 2's reduction (via the Loewner form of DS11(a)) identifies the degenerate attainers, but the uniqueness/rigidity theory of the vector between-matrix problem is open — the dichotomy transfers exactly when that theory does. (ii′) $d_\lambda\ge2$: at $K=d_\lambda+1$ the claim is false (rank vacuity, CE-DS-MARGINS-RANK-VACUITY-001); at $K\ge d_\lambda+2$ it is open — Proposition 6's steering uses a scalar nuisance sign and (R) as stated is one-dimensional (a vector version needs sign-rich swap directions spanning $\mathbb R^{d_\lambda}$ and $K\ge d_\lambda+2$ for the constraint-plane target to exist); tracked in OPEN-DS-MARGINS-NONCENTERED. (iii) Local/ exchange-stable non-global sequences: DS14 continues to govern them under certified margins; nothing here says local optima degenerate. (iv) Estimated scores: “Estimated scores and classifiers”. (v) (M5): at optima the projected-centroid object $\hat e_b=\hat\mu_{\psi,b}-\hat B_z\hat\mu_{\lambda,b}$ rides on $\hat B_z=\hat I_{\psi\lambda}/\hat I_{\lambda\lambda}\to0/0$; the meaningful reduced geometry is the scalar $J^*$ geometry (distinct centroids). Measured: the optimum's own regression slope keeps an $O(1)$-scale distance from the full-sample slope with no decline through $N=20$ (median per-law gaps $0.01$–$0.6$) — the (M5) object is indeed unstable at optima.

Measured (N-DS-MARGINS-TREND, N-DS-SCALAR-MASS, N-DS-MARGINS-EXACT-ANCHOR). Fine-grid ($1/2^{16}$, atomless-emulating) exact-optimum suite, 6 laws, $N=8$–$18$, 3 reps (+ $N=20$ extension): the sandwich and tax identity verified exactly on every instance (112/112, including four at $N=20$); zero exact ties (the 31-fold tie was an atomic-grid artifact); the optimum is always within 0–3 points of the best efficient-score interval labeling, with relative value gap medians 0.3–6%; gauss06's binned nuisance block declines (median 0.23 at $N=8$ to 0.02 at $N=18$, the predicted $K/N$ scale) while the non-centered mix3's stays macroscopic (median 0.30–0.68 through $N=16$) — the class boundary is visible. Singletons still occur at these tiny $N$ but thin out for the Gaussian laws; the scalar suite settles their fate: exact optimal $K$-interval partitions of Gaussian/Laplace/uniform samples at $N=100$–$20{,}000$ (12 reps each) have min cell mass rising to the population values (Gaussian $K=3$: 0.259–0.268 at $N=20{,}000$ vs 0.2703; $K=6$: 0.063–0.071 vs 0.0740) with no singleton beyond $N=100$. The float screen is anchored by fully exact enumeration (all 86,526 canonical partitions per instance) at $N=12$: 6/6 optima match exactly.

Audit-side measured (30 Aug 2026, N-DS-AUDIT15 rows). An independent pure-stdlib exact suite (py/audit_ds_margins_at_optima.py, own CLT-rational law emulation and LCG seeds, no float screen) exhaustively certified 20 exact global optima at $N=12$–$16$ (up to all 7,141,686 canonical partitions per instance): the sandwich and tax identity hold on 20/20; zero exact ties over the full lattice on every instance; the researcher's float top-64 screen, re-implemented with its $10^{-9}$ guard, ranks the true optimum first with zero guard casualties on 20/20 — validating the screen mechanism in the $N\ge14$ territory where the original suite is screen-selected (the original $N\ge14$ instances themselves remain uncertified); the centered law's nuisance block stays small while the non-centered control's stays macroscopic (class boundary reproduced); the single observed singleton sits on the non-centered control ($N=15$). The float-only scalar (M2) sweep is anchored by an exact-rational interval DP at $N=1000$: the library DP reproduces the exact optimum (SSE agreement $10^{-9}$, identical min cell mass $49/200$).

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