Margined profiled stable labellings become geometric in the limit

Let $S_1,\dots,S_N$ be i.i.d. from $P$ with equal weights, and let $z^{(N)}$ be one-point exchange-stable finite $D_s$ labelings into $K$ cells. Write $\hat I_N$, $\hat\mu_b$, $\hat G_s$, $\hat e_b$ for the labeling's own binned quantities and let the companion rule $\rho_N$ be the efficient-semimetric nearest-cell rule built from them ($\rho_N(s)=\arg\min_b(\hat e(s)-\hat e_b)^\top S_\psi(\hat I_N)^{-1}(\hat e(s)-\hat e_b)$).

Assumptions.

• (M1) $P$ atomless, $E[S]=0$, $E\|S\|^2<\infty$;
• (M2) mass margin: $\min_b\hat W_b\ge c_0>0$;
• (M3) conditioning margin: $\lambda_{\min}(\hat I_N)\ge\kappa>0$;
• (M4) slab margin: $\sup_{\|v\|=1,c}P(|v^\top S-c|\le t)\le\varphi(t)$, $\varphi(t)\downarrow0$;
• (M5) projected-centroid separation: $\min_{b\ne b'}\|\hat e_b-\hat e_{b'}\|\ge\gamma>0$;

(M2)/(M3)/(M5) along the sequence, almost surely eventually.

Theorem. Almost surely:

1. (Geometrization.) $P_N(z^{(N)}\ne\rho_N)\to0$: the labeling disagrees with its own companion rule on a vanishing fraction of the sample.
2. (Subsequential population stationarity.) Along any subsequence with converging rule parameters, $\rho_N\to q^*$ $P$-a.e., where $q^*$ is a self-consistent population efficient-Voronoi quantizer — by DS12 exactly a bounded-study stationary population quantizer — and $\hat I_N(z^{(N)})\to I_{q^*}$, $\hat\Phi_s(z^{(N)})\to\Phi_s^{\rm pop}(q^*)$.
3. (Global variant.) If each $z^{(N)}$ is a global finite $D_s$ optimum, then $\hat\Phi_s(z^{(N)})\to v^*=\sup\{\Phi_s^{\rm pop}(\rho)\}$ over the compact class of efficient-Voronoi rules compatible with the margins $(c_0,\kappa,\gamma)$, and every subsequential limit $q^*$ attains $v^*$.

Proof.

Step 1 (finite near-geometry). By DS13, every misassigned point (positive rule violation $g_i=s_{aa}-s_{b^*b^*}>0$) satisfies $g_i\le\frac1N q_{aa}q_{b^*b^*}$. Under (M2)–(M3), $\|\hat\mu_b\|^2\le M_N/c_0$ with $M_N=P_N\|S\|^2\to M=E\|S\|^2$, so $q_{xx}\le\frac2\kappa(\|s_i\|^2+M_N/c_0)=:Q_i$ and $g_i\le Q_i^2/N$. Hence for any $t>0$, a misassigned point has either violation gap $\le t$ or $\|s_i\|^2\ge\sqrt{tN}\,\kappa/2-M_N/c_0$; Markov's inequality gives

$$P_N(z^{(N)}\ne\rho_N) \le P_N(0<\mathrm{gap}\le t)+ \frac{M_N}{\sqrt{tN}\,\kappa/2-M_N/c_0}.$$

Step 2 (band mass). Pairwise decision functions $h_{bb'}(s)$ are affine in $s$ with normals $v_{bb'}=2C_N^\top S_\psi(\hat I_N)^{-1}(\hat e_b-\hat e_{b'})$; since $\|C_N^\top w\|\ge\|w\|$ and $\kappa\preceq S_\psi(\hat I_N)\preceq\Lambda$ — where $\Lambda:=2M$ is derived, not assumed: $\lambda_{\max}(S_\psi(\hat I_N))\le\operatorname{tr}\hat I_N =\sum_b\hat W_b\|\hat\mu_b\|^2\le M_N\to M$ a.s., and the lower bound is $S_\psi=((\hat I_N^{-1})_{\psi\psi})^{-1}\succeq\lambda_{\min}(\hat I_N)$ (audit §8) — (M5) gives $\|v_{bb'}\|\ge2\gamma/\Lambda$. The empirical gap-$t$ band, whatever its data-dependent normals, lies in $\binom K2$ members of the fixed family $\mathcal S=\{\{s:|v^\top s-c|\le r\}\}$ of all slabs of half-width $r=t\Lambda/(2\gamma)$; $\mathcal S$ is a VC class (intersections of two half-spaces), so $\sup_{\mathcal S}|P_N-P|\to0$ a.s. (VC Glivenko–Cantelli; van der Vaart–Wellner Thm 2.4.3 / Pollard 1984), and (M4) — already uniform over all $(v,c)$ — bounds every population slab mass, giving $\limsup_NP_N(0<\mathrm{gap}\le t)\le\binom K2\varphi(t\Lambda/(2\gamma))$. Sending $N\to\infty$ then $t\downarrow0$ in Step 1 proves conclusion 1. (Audit: the uniform law must run over the fixed class, never the data-dependent slabs; this is the repair recorded in §8 of the audit.)

Step 3 (moment identification). $|\hat W_b(z^{(N)})-\hat W_b(\rho_N)|\le P_N(z\ne\rho)\to0$ and $\|\hat m_b(z^{(N)})-\hat m_b(\rho_N)\|\le M_N^{1/2}P_N(z\ne\rho)^{1/2}\to0$ (Cauchy–Schwarz). The companion rules lie in the compact affine-max class with parameters bounded by $(c_0,\kappa,\gamma,M)$; by the C1 uniform moment convergence over that class, $\sup_\rho\|\hat m_b(\rho)-m_b^P(\rho)\|\to0$ and likewise for masses. Along a subsequence with converging parameters, dominated convergence (tie sets are $P$-null by (M4)–(M5)) gives $\rho_N\to q^*$ a.e. and population moments converge; chaining the three approximations identifies $\lim\hat W_b(z^{(N)})=W_b^P(q^*)$, $\lim\hat\mu_b(z^{(N)})=\mu_b^P(q^*)$, $\lim\hat I_N=I_{q^*}\succeq\kappa$.

Step 4 (self-consistency and stationarity). The rule $q^*$ is built from the limit centroids and the limit metric, which Step 3 identifies as the population centroids and metric of $q^*$ itself; hence $q^*$ is a self-consistent efficient-Voronoi quantizer, and by the DS12 equivalence it is bounded-study stationary. Continuity of $F$ at $I_{q^*}\succ0$ gives $\hat\Phi_s(z^{(N)})=F(\hat I_N)\to F(I_{q^*})$ — conclusion 2.

Step 5 (global sandwich). For any fixed margin-compatible rule $\rho$, the sample labeling induced by $\rho$ is feasible (nonempty cells a.s. eventually), so $\hat\Phi_s(z^{(N)})\ge\hat\Phi_s(\rho\text{-labels})\to \Phi_s^{\rm pop}(\rho)$; hence $\liminf\hat\Phi_s(z^{(N)})\ge v^*$. Along any parameter-convergent subsequence, Step 4 gives $\hat\Phi_s(z^{(N)})\to\Phi_s^{\rm pop}(q^*)\le v^*$ because $q^*$ inherits the margins. Both bounds force $\lim\hat\Phi_s(z^{(N)})=v^*=\Phi_s^{\rm pop}(q^*)$. ∎

Merged-rule variant (dropping (M5)). Without a separation margin, pass to a subsequence along which every pairwise separation converges; merge cell pairs whose separation vanishes (a genuine equivalence relation by the triangle inequality). Steps 1–2 apply verbatim to the merged companion rule (a group-level misassignment is in particular a cross-group cell-level one, so DS13 supplies the same gap bound; only inter-group boundaries carry slabs, and their normals are bounded below along the subsequence), and Steps 3–5 deliver a population-stationary reduced rule (DS11(d), DS12). The value identification uses the general-$d_\psi$ form of DS11(b): within a group the fine rule's own $B^*(\hat I_N)$ equalizes projected centroids in the limit, so the limit fine and merged configurations have exactly equal profiled information (audit §8). This is sharp: CE-DS-DEGENERATE-GLOBAL-TIE-001 shows exact finite global optima whose label-level structure is not identified (31-fold exact tie), while the reduced configuration is unique.

Audit (28 Aug 2026). The theorem passed the publication-grade audit (AUDITS/AUDIT-DS-POPULATION-BRIDGE-001.md) as a conditional result: all five steps were independently re-derived, with the fixed-class slab Glivenko–Cantelli argument, the derived $\Lambda$, the explicit compact parameter set for Steps 3/5, the two-sided Step-4 fixed-point identification, and the merged-variant value argument supplied. The audit also notes a free strengthening: in conclusion 3 the comparison class may be broadened to all geometric rules with positive masses, distinct centroids, and nonsingular information — the margins bind only through the hypotheses on $z^{(N)}$.

What is deliberately not claimed. (i) The margins (M2)/(M3)/(M5) are not automatic at finite optima — and DS15 (29 Aug 2026) settles how, for conditionally centered laws at $d_\psi=1$: (M2) is automatic asymptotically (the $N\le18$ singleton evidence is pre-asymptotic), but (M3) provably fails at free global optima, whose limits are nuisance-degenerate efficient-score interval quantizers. On that class DS14 therefore governs margin-certified solutions — necessarily suboptimal by $\delta(\kappa)=v_K-v^*(\kappa)>0$ — never free global optima; beyond the class the margins remain open (OPEN-DS-MARGINS-NONCENTERED). (ii) $v^*$ is the optimum over the margin-compatible geometric class; DS15 shows it sits strictly below the unrestricted population supremum $v_K$ on the conditionally centered class, where $v_K$ is attained only degenerately; the general attainment question remains C2. (iii) Everything is for exact scores; estimated-score robustness is the “Estimated scores and classifiers” theme.

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