Population stationarity can defeat a geometric rule

Let $P$ be atomless with $E[S]=0$, $E\|S\|^2<\infty$, and let $q$ have $W_b>0$ for all $b$ and $I_q\succ0$. Call $q$ bounded-study stationary if for every $a\ne b$ and every $R>0$,

$$\limsup_{\substack{E\subseteq A_a\cap B(0,R)\\ P(E)\to0}} \frac{\Phi_{D_s}(q_{E\to b})-\Phi_{D_s}(q)}{P(E)}\le0,$$

where $q_{E\to b}$ relabels the measurable set $E$ to cell $b$. (Any local maximizer over small-mass bounded relocations is bounded-study stationary.)

Study dictionary. $I_q$ depends on the partition only through $(W_b,m_b)$, so relabeling a set $E$ with mass $\varepsilon=P(E)$ and barycenter $\bar s=\frac1\varepsilon\int_ES\,dP$ changes $I_q$ exactly as the finite rank-two relocation D2 of the weighted point $(\bar s,\varepsilon)$: $\Delta I=\alpha u_au_a^\top-\beta u_bu_b^\top$, $u_x=\bar s-\mu_x$, $\alpha=\varepsilon W_a/(W_a-\varepsilon)$, $\beta=\varepsilon W_b/(W_b+\varepsilon)$. All finite relocation algebra transfers verbatim to population studies.

Theorem. $q$ is bounded-study stationary iff for every $a$, $P$-a.e. $s\in A_a$ and every $b$:

$$\boxed{ (s-\mu_a)^\top G_s(s-\mu_a)\le(s-\mu_b)^\top G_s(s-\mu_b), \qquad G_s=I_q^{-1}-E_\lambda(I_q)_{\lambda\lambda}^{-1}E_\lambda^\top . }$$

Moreover $G_s=C^\top S_\psi(I_q)^{-1}C$ with $C=[\,\mathrm{Id}_{d_\psi}, -B^*_q\,]$, so the rule reads: assign a.e. to the nearest projected centroid $e_b=C\mu_b$ of the efficient projection $e(s)=Cs$ in the $S_\psi(I_q)^{-1}$ metric,

$$q(s)\in\arg\min_b\,(e(s)-e_b)^\top S_\psi(I_q)^{-1}(e(s)-e_b) \quad\text{a.e.}$$

Proof. The pairwise first-variation function $\delta_{ab}(s)=(s-\mu_a)^\top G_s(s-\mu_a)-(s-\mu_b)^\top G_s(s-\mu_b) =2(\mu_b-\mu_a)^\top G_ss+\mu_a^\top G_s\mu_a-\mu_b^\top G_s\mu_b$ is affine in $s$ (the quadratic terms cancel; this is G1's common-$G$ affinity). $\Phi_{D_s}=F(I)$ with $F=\log\det S_\psi$ is $C^2$ on a neighborhood of $I_q\succ0$ with $\nabla F(I_q)=G_s$ (DS2), so for a study with bounded barycenter the exact update gives $\Phi(q_{E\to b})-\Phi(q)=P(E)\,\delta_{ab}(\bar s)+O(P(E)^2)$, with the remainder uniform over $\bar s\in B(0,R)$.

(⟹) If $V=\{s\in A_a:\delta_{ab}(s)\ge c\}\cap B(0,R)$ had $P(V)>0$ for some $c>0$, atomlessness supplies studies $E_n\subseteq V$ with $P(E_n)\to0$; affinity gives $\delta_{ab}(\bar s_n)\ge c$ (the barycenter average of an affine function), so the gain per unit mass is at least $c-O(P(E_n))>0$, contradicting stationarity.

(⟸) For any study $E\subseteq A_a\cap B(0,R)$, affinity gives $\delta_{ab}(\bar s_E)=\frac1{P(E)}\int_E\delta_{ab}\,dP\le0$, so the gain is $\le O(P(E)^2)$ and the limsup is $\le0$. ∎

The factorization $G_s=C^\top S_\psi^{-1}C$ follows from the block inverse: $(I^{-1})_{\psi\psi}=S_\psi^{-1}$, $(I^{-1})_{\psi\lambda}=-S_\psi^{-1}B^*_q$, $(I^{-1})_{\lambda\lambda}=I_{\lambda\lambda}^{-1}+B_q^{*\top}S_\psi^{-1}B^*_q$.

Deployability characterization (an earlier, since-closed question). The nearest-projected-centroid correspondence is $P$-a.e. single-valued and reproduces $q$ up to null sets iff (i) the projected centroids $e_b$ are pairwise distinct and (ii) $P$ charges no pairwise tie hyperplane (each tie set is affine in $e(s)$; both conditions hold e.g. when $P$ is absolutely continuous and (i) holds). (Audit hardening: the selection-independent reading is the correct one — a tie-breaking selection could reproduce $q$ even when (ii) fails, so the characterization is about the rule being determined by the geometry alone; audit §8. Note also the asymmetry: sufficiency (⟸) of the stationarity theorem above holds for every $P$, while necessity (⟹) genuinely needs atomlessness — for a finitely atomic law bounded-study stationarity is vacuous and CE-DS-GLOBAL-GEOMETRY-001 violates the rule.) Stationarity does not force (i): CE-DS-POP-WASTED-CELLS-001 is a stationary partition (zero violations, ties allowed) whose coincident-centroid cell pairs no efficient-semimetric rule can separate — in sharp contrast to finite D, where exchange stability forces distinct centroids (D5). Without (i), stationarity constrains exactly the reduced partition obtained by merging coincident-$e_b$ groups (cf. DS11(d)).