Global profiled optima can have degenerate ties

Audit (28 Aug 2026, AUDITS/AUDIT-DS-POPULATION-BRIDGE-001.md): the boxed identity is classical — it is the extremal characterization of the generalized Schur complement (Krein 1947; Anderson's shorted operator 1971; Li–Mathias, SIAM Review 42(2), 2000, Thm 2.2, including the Loewner order, the pseudo-inverse, and the exact attainment set). The statistical reading $V(B)=\operatorname{Var}(E[S_\psi-BS_\lambda\mid Z])$ additionally requires centered scores ($E[S]=0$). The binned transfer and consequences (a)–(d) are project-level.

For any partition with binned information blocks $I_{\psi\psi},I_{\psi\lambda},I_{\lambda\lambda}$ define the generalized profiled information $S_\psi^+(I)=I_{\psi\psi}-I_{\psi\lambda}I_{\lambda\lambda}^+I_{\lambda\psi}$ (Moore–Penrose pseudo-inverse; the ordinary Schur complement when $I_{\lambda\lambda}\succ0$). Then

$$\boxed{ S_\psi^+(I_q) =\min_B\operatorname{Var}\!\bigl(E[S_\psi-BS_\lambda\mid Z]\bigr) =\min_B\sum_bW_b(\mu_{b\psi}-B\mu_{b\lambda})(\mu_{b\psi}-B\mu_{b\lambda})^\top, }$$

a Loewner minimum over $d_\psi\times d_\lambda$ matrices $B$, attained at every solution of $BI_{\lambda\lambda}=I_{\psi\lambda}$ (in particular $B_q^*=I_{\psi\lambda}I_{\lambda\lambda}^+$; solutions exist because $\operatorname{range}(I_{\lambda\psi})\subseteq\operatorname{range}(I_{\lambda\lambda})$ for a PSD block matrix).

Proof. $V(B)=I_{\psi\psi}-BI_{\lambda\psi}-I_{\psi\lambda}B^\top +BI_{\lambda\lambda}B^\top$; for any normal-equation solution $B_0$, completion of squares gives $V(B)=V(B_0)+(B-B_0)I_{\lambda\lambda}(B-B_0)^\top\succeq V(B_0)$, and $V(B_0)=S_\psi^+(I)$. ∎

Consequences.

(a) Efficient-score domination, singular extension, and the exact equality condition (an earlier, since-closed question, fixed $q$). Evaluating at $B=B^*_{\rm full}$ re-derives DS7 in one line, extends it to singular binned nuisance blocks, and gives the exact gap

$$\boxed{ \operatorname{Var}(E[\widehat S\mid q])-S_\psi^+(I_q) =(B^*_{\rm full}-B^*_q)\,I^q_{\lambda\lambda}\,(B^*_{\rm full}-B^*_q)^\top \succeq0 . }$$

Equality for a fixed $q$ holds iff $(B^*_{\rm full}-B^*_q)I^q_{\lambda\lambda}=0$: the full-data nuisance projection is already optimal for the binned centroids. The finite-$K$ gap is therefore exactly the cost of estimating the nuisance projection from bins. Along any refining sequence of partitions generating the Borel $\sigma$-field, $E[S\mid Z_K]\to S$ in $L^2$ (Lévy upward martingale convergence), so $I_{q_K}\to I_{\rm full}$; provided $I^{\rm full}_{\lambda\lambda}\succ0$ (implicit in defining $\widehat S$), the binned nuisance blocks are eventually nonsingular, where $B\mapsto I_{\psi\lambda}I_{\lambda\lambda}^{-1}$ is continuous, so $B^*_{q_K}\to B^*_{\rm full}$ and the gap vanishes — the $K\to\infty$ part of an earlier, since-closed question. (The pseudo-inverse map is discontinuous at rank drops — witness $I_{\lambda\lambda}^{(k)}=\mathrm{diag}(1,1/k)$, $I_{\psi\lambda}=[1,1]$, $B^*_k=[1,k]$ — so the nonsingular-limit hypothesis is load-bearing; audit §8.) The behavior at global optima was an earlier, since-closed question and is now settled by DS15 for conditionally centered laws at $d_\psi=d_\lambda=1$: the gap is the projection tax and vanishes a.s. along free global optima (DS15 conclusion 5); beyond that class it is OPEN-DS-MARGINS-NONCENTERED.

(b) Refinement monotonicity with exact equality characterization ($d_\psi=1$). Splitting cell $M$ into $(x,y)$: for every slope $c$, with $e_b(c)=\mu_{b\psi}-c\,\mu_{b\lambda}$,

$$V(c;\text{split})=V(c;\text{merged}) +\frac{W_xW_y}{W_M}\bigl(e_x(c)-e_y(c)\bigr)^2 .$$

Hence $S_\psi^+$ never decreases under refinement, and the split is profiled-information-neutral iff some $c$ simultaneously minimizes $V(\cdot;\text{merged})$ and equalizes $e_x(c)=e_y(c)$. (Audit: the identity is the exact between-group variance decomposition; the "iff" holds by evaluating at a minimizer of the split problem, which exists by the normal-equation argument. The same statement and proof hold for every $d_\psi\ge1$ with the rank-one PSD gap $\frac{W_xW_y}{W_M}(e_x(B)-e_y(B))(e_x(B)-e_y(B))^\top$ and Loewner sandwiching — the form DS14's merged variant consumes; audit §8.)

(c) Wasted cells and exact global ties. If the merged configuration is entirely nuisance-degenerate (all cells have $\mu_{b\lambda}=0$, so $V(\cdot;\text{merged})$ is constant and every slope minimizes it), every split whose sub-cells have distinct nuisance means is exactly neutral, while a split with equal (hence zero) nuisance means and distinct POI means strictly increases $S_\psi^+$ but keeps the nuisance block singular — it leaves the in-bin formulation (DS9). (Audit hardening: the neutrality argument needs the minimizer set of $V(\cdot;\text{merged})$ to meet the equalizing slopes; full nuisance degeneracy supplies that, a partially degenerate configuration need not.) Exact witnesses:

• COUNTEREXAMPLES/CE-DS-DEGENERATE-GLOBAL-TIE-001.json: a centered equal-weight $N=8,d=2,d_\psi=1,K=3$ sample whose exact global in-bin optimum $1083/4096$ is attained by 31 distinct labelings — exactly the feasible refinements of one reduced bipartition — every one of which has two exactly coincident projected centroids; the unique nuisance-mean-equal refinement is infeasible (singular nuisance block) with generalized value $1191/4096>1083/4096$.
• COUNTEREXAMPLES/CE-DS-POP-WASTED-CELLS-001.json: the population/quadrature wasted-cell construction of DS12.

(d) Identifiability up to neutral splits. The profiled objective is invariant under $\Phi$-neutral splits, so a finite global optimum is in general identified only up to the reduced configuration of projected centroids $\{(W_b,e_b(B^*_q))\}$; deployable content lives at the reduced level.