One explicit non-centered law has convergent global profiled optima

Independent audit (31 Aug 2026, AUDITS/AUDIT-DS-NONCENTERED-GLOBAL-BASIN-TRANSFER-001.md). Verified with hardened assumptions. The population optimum, its uniqueness, the strict isolation, the almost-sure transfer of every global-optimizer sequence, the finite exchange stability and the fixed margins all survive independent adversarial attack, with 65,924 exactly enumerated canonical partitions and a two-route recomputation of the law. Five hardenings, all folded into the text below. (H1) The proof no longer cites DS15 Lemma 3 or Proposition 5: both are registered for class (L) — which this law violates by construction — under a chapter convention of exactly centered empirical scores, which DS18.2 explicitly negates. The scalar steps are now proved here for this law. (H2) The ordinary comparison domain is the in-bin (DS9) one; a zero binned nuisance block is shown to be null under the law, so the regularity restriction is a.s. vacuous, while CE-DS-NONCENTERED-SINGULAR-DESTINATION-001 shows the convention is load-bearing off that null set. (H3) Population uniqueness is a.s. up to labels and null sets. (H4) The probability-one event is exhibited explicitly and the transfer is given a computable finite-$N$ rate. (H5) (M3) is $\lambda_{\min}(\hat I_N)\ge\kappa$, not $\det/\operatorname{tr}$ — under the latter reading the registered $\kappa=1/4$ would fail, since $\det I_{q^*}/\operatorname{tr}I_{q^*}=32/189$. The audit also repaired two prior-art defects: six direct antecedents already in the project bibliography were unlinked, and the uniqueness citation was mis-scoped to a strict-log-concavity result that $\operatorname{Unif}[-1,1]$ fails.

DS17 left two logically separate gaps off class (L): exhibit a regular root with fixed margins, and show that an empirical sequence can inhabit it despite boundary-scale one-point gains. Both gaps are resolved here for one explicit bounded score law. The transfer is through finite global profiled optima; it is not a theorem that raw population labels are finite terminals, nor a selection theorem for exchange ascent.

The named law and exact population rule

Let $X,Z\stackrel{\mathrm{iid}}\sim\operatorname{Unif}[-1,1]$ and define

$$S_\psi=X,\qquad S_\lambda=3X^2-1+Z.$$

Then $ES=0$, $E\|S\|^2<\infty$, and

$$I_{\rm full}=E[SS^\top] =\begin{pmatrix}1/3&0\\0&17/15\end{pmatrix}\succ0.$$

The full-data regression slope is $B^*=0$, hence the efficient score is $\hat s=X$. The law is strictly outside (L), because

$$E[S_\lambda\mid\hat s=X]=3X^2-1\ne0\quad\text{a.s.}$$

It is atomless and satisfies (M4). Indeed its density is (1/4) on

$$\{(x,y):|x|\le1,\ |y-(3x^2-1)|\le1\} \subset[-1,1]\times[-2,3].$$

For every unit $v$, a width-$2t$ slab intersects the bounding rectangle in area at most $2t\sqrt{29}$; therefore one may take $\varphi(t)=\min(1,\sqrt{29}t/2)\downarrow0$.

Let $q^*$ be the three-cell $X$-interval rule with cuts $(-1/3,1/3)$. Direct integration gives

$$W_b=1/3,\qquad \mu_{\psi,b}=(-2/3,0,2/3),\qquad \mu_{\lambda,b}=(4/9,-8/9,4/9),$$

and hence

$$I_{q^*} =\begin{pmatrix}8/27&0\\0&32/81\end{pmatrix},\qquad \Phi_{D_s}(q^*)=8/27.$$

At $\beta=0$, the projected cell means are $(-2/3,0,2/3)$, whose midpoints are exactly the declared cuts. Thus $q^*$ is Lloyd-stationary for $T_0=X$, and the DS17.4 residual is

$$E[h(X)S_\lambda]=I_{\psi\lambda}(q^*)=0.$$

Since $I_{\lambda\lambda}(q^*)=32/81>0$, this is a regular root with $B^*_{q^*}=0$, not a singular numerator root. Its exact gate margins are

$$\min_bW_b=1/3,\qquad \lambda_{\min}(I_{q^*})=8/27,\qquad \min_{b\ne b'}|t_b-t_{b'}|=2/3.$$

In particular the fixed constants $(c_0,\kappa,\gamma)=(1/4,1/4,1/2)$ have strict slack. The root also obeys the DS17 compact tilt bound trivially because $\beta=0$. Relative to the unbinned profiled information $1/3$, its scalar $D_s$ retention is

$$\eta_{D_s}=\frac{8/27}{1/3}=\frac89.$$

Theorem DS18.1 (unique strict population attainer)

Among every measurable regular three-cell quantizer of $S$, $q^*$ is the unique population $D_s$ maximizer up to cell relabeling. Moreover it is strictly isolated in decision distance: for every $\varepsilon>0$ there is $\delta(\varepsilon)>0$ such that

$$\min_{\pi}\sum_bP(A_b\mathbin\triangle A^*_{\pi(b)})\ge\varepsilon \quad\Longrightarrow\quad \Phi_{D_s}(q)\le 8/27-\delta(\varepsilon).$$

Proof (audit form; self-contained). At $d_\lambda=1$ the first inequality needs no variational machinery: $\Phi_{D_s}(q)=I_{\psi\psi}(q)-I_{\psi\lambda}(q)^2/I_{\lambda\lambda}(q)\le I_{\psi\psi}(q)$, and at a singular block $I_{\psi\lambda}(q)=0$ too, so the DS11 pseudo-inverse extension gives equality — the bound holds under both conventions. The second is the nearest-codepoint reduction, which uses only the pointwise inequality $(X-m_{q(S)})^2\ge\min_b(X-m_b)^2$ with $m_b=E[X\mid q=b]$ and therefore holds for arbitrary measurable cells, including cells that depend on $Z$ and are not $X$-measurable:

$$\Phi_{D_s}(q) \le \sum_bW_bE[X\mid q=b]^2 = E[X^2]-E[(X-m_q)^2] \le v_3(\operatorname{Unif}[-1,1]).$$

For a uniform interval of length $\ell$, the within-cell squared-error contribution (including density $1/2$) is $\ell^3/24$. The three interval lengths sum to $2$, so strict convexity gives the unique minimum at $\ell_1=\ell_2=\ell_3=2/3$. Its total error is $1/27$, and therefore

$$v_3=E[X^2]-1/27=1/3-1/27=8/27.$$

The exact calculation above shows $q^*$ attains both bounds with a regular nuisance block. Equality forces the centroid set to be an optimal codebook, hence $\{-2/3,0,2/3\}$ — in particular three distinct centroids — and then forces $m_{q(S)}$ to be a nearest codepoint a.s.; the uniform law charges neither midpoint $\pm1/3$, so $q=q^*$ a.s. up to labels and null sets (H3). A cell of zero mass is excluded because $v_2=1/3-1/12=1/4<8/27$, and a singular $q$ is excluded because the unique attainer $q^*$ has $I_{\lambda\lambda}=32/81>0$; regularity is therefore not load-bearing for uniqueness.

The quantitative isolation is proved directly (H1), not imported. Let $G(c)=E[\min_b(X-c_b)^2]-1/27\ge0$ on $[-1,1]^3$; it is continuous and vanishes only at $c^*$, so by compactness $g(\rho)=\inf\{G(c):\|c-c^*\|_\infty\ge\rho\}>0$ for every $\rho>0$. If $\Phi_{D_s}(q)\ge8/27-\delta$ then $G(m)\le\delta$, so $\|m-c^*\|_\infty<\rho$ once $\delta<g(\rho)$. Taking $\rho=\eta/8$, every $x$ with $|x\mp1/3|>\eta$ keeps a margin $>\tfrac43\eta-8\rho=\tfrac13\eta$ for its correct codepoint, so $P(q\ne q^*,|X\mp1/3|>\eta)\le3\delta/\eta$ while the two boundary bands carry mass $2\eta$. Hence $d(q,q^*)\le2(3\delta/\eta+2\eta)$ whenever $\delta<g(\eta/8)$, which is the claimed $\delta(\varepsilon)$ in contrapositive. The local face of $g$ is exact: the distortion Hessian at $c^*$ is $\bigl[\begin{smallmatrix}1/2&-1/6&0\\-1/6&1/3&-1/6\\0&-1/6&1/2\end{smallmatrix}\bigr]$ with $\lambda_{\min}=1/6$ — a fact the cited Liu-Pages-2020 route does not supply here, since its Proposition 11 assumes strict log-concavity. Scalar uniqueness itself is classical for log-concave densities (Kieffer-1983, Mease-Nair-2006). ∎

Theorem DS18.2 (almost-sure exact empirical inhabitation)

Let $S_1,S_2,\ldots$ be i.i.d. from the named law, with equal weights $1/N$ and without sample centering. For every sufficiently large $N$, let $z^{(N)}$ be any exact global maximizer of ordinary in-bin profiled $D_s$ over the finite labelings with three nonempty cells and $\hat I^z_{\lambda\lambda}>0$. Then, on one probability-one event, after relabeling,

$$P_N(z^{(N)}\ne q^*)\to0,\qquad \hat I_N(z^{(N)})\to I_{q^*},\qquad \hat\Phi_{D_s}(z^{(N)})\to8/27.$$

Consequently all sufficiently large $z^{(N)}$ are exact ordinary one-point exchange-stable and satisfy (M2)+(M3)+(M5) at the fixed constants $(1/4,1/4,1/2)$. Their companion slopes tend to $0$, their projected centroids tend to $(-2/3,0,2/3)$, and their companion rules converge to the deployable population rule $q^*$.

Proof (audit form). The probability-one event is exhibited explicitly (H4) and is selection-independent — none of its four members mentions a labeling, which is exactly what licenses the "every sequence" quantifier: $\Omega_0=\Omega_1\cap\Omega_2\cap\Omega_3\cap\Omega_4$ with $\Omega_1=\{\sup_{c\in[-1,1]^3}|P_Nf_c-Pf_c|\to0\}$ for $f_c(x)=\min_b(x-c_b)^2$ (bounded by $4$ and $4$-Lipschitz in $c$ on the cube, so a finite net plus the SLLN suffices); $\Omega_2$ the SLLN on the finitely many fixed functionals $1_{A^*_b},X1_{A^*_b},S_\lambda1_{A^*_b},X^2$; $\Omega_3=\{P_N(|X\mp1/3|\le\eta)\to\eta\ \forall\eta\in\mathbb Q\cap(0,1/3)\}$; and $\Omega_4=\{\sum_{i\le N}S_{\lambda,i}\ne0\ \forall N\}$. The last is where regularity is settled: at $d_\lambda=1$, $\hat I_{\lambda\lambda}=0$ forces every cell $\lambda$-sum, hence the total, to vanish, and $\sum_iS_{\lambda,i}$ is absolutely continuous, so on $\Omega_4$ every three-nonempty-cell labeling is regular (H2).

Label the sample by the fixed cuts of $q^*$ — half-open, $\{X<-1/3\},\{-1/3\le X<1/3\},\{X\ge1/3\}$ — writing the result as $z_N^*$. The SLLN on these three fixed cells gives $\hat I_N(z_N^*)\to I_{q^*}$, so $z_N^*$ is feasible eventually and $\hat\Phi_{D_s}(z_N^*)\to8/27$.

For any finite labeling $z$, the same variational evaluation at $B=0$ and scalar reassignment used above are exact sample algebra:

$$\hat\Phi_{D_s}(z) \le \operatorname{btw}_N(X;z) \le \hat v_{3,N}(X),$$

where — uncentered, about the origin, since the sample is never centered — $\hat v_{3,N}(X)=P_N[X^2]-\hat D_{3,N}$ with $\hat D_{3,N}=\min_cP_N\min_b(X-c_b)^2$, attained by intervals of the sorted $X_i$. On $\Omega_1$, $\hat D_{3,N}\to1/27$ and $P_N[X^2]\to1/3$, so $\hat v_{3,N}(X)\to8/27$ a.s. (H1: this replaces the citation to DS15 Proposition 5, whose statement carries an estimated tilt and an exact-centering shift, neither of which exists here). Global optimality therefore squeezes

$$8/27\leftarrow\hat\Phi_{D_s}(z_N^*) \le\hat\Phi_{D_s}(z^{(N)}) \le\operatorname{btw}_N(X;z^{(N)}) \le\hat v_{3,N}(X)\to8/27.$$

Rigidity is then quantitative rather than asymptotic (H4). Put

$$\Delta_N:=\hat v_{3,N}-\hat\Phi_{D_s}(z_N^*)\ \ge0 ,$$

computable from the sample with no search over labelings. Every labeling with $\hat\Phi_{D_s}(z)\ge\hat\Phi_{D_s}(z_N^*)$ — in particular every global optimum — has own-codebook excess $\le\Delta_N$ and a $\Delta_N$-optimal codebook, so on $\Omega_1$ its centroids converge to $(-2/3,0,2/3)$ and the §DS18.1 margin argument transfers verbatim to $P_N$:

$$P_N\bigl(z^{(N)}\ne q^*\bigr) \;\le\;\frac{3\Delta_N}{\eta}+P_N\bigl(|X-\tfrac13|\le\eta\bigr)+P_N\bigl(|X+\tfrac13|\le\eta\bigr)$$

for every rational $\eta\in(0,1/3)$ and all large $N$; letting $N\to\infty$ on $\Omega_3$ and then $\eta\downarrow0$ gives $P_N(z^{(N)}\ne q^*)\to0$ after relabeling. Because $|S_\psi|\le1$ and $|S_\lambda|\le3$, uniform integrability is free and the fixed-rule SLLN plus this vanishing disagreement gives convergence of every cell mass and score moment, hence of the information, slope, projected centroids, and objective. The strict limiting margins $(1/3,\,8/27,\,2/3)$ admit the displayed rational lower constants — with (M3) read as $\lambda_{\min}(\hat I_N)\ge\kappa$ (H5).

Finally, an exact global feasible labeling has no admissible improving one-point relocation, so it is exact ordinary exchange-stable — at every $N$ on $\Omega_4$, not merely eventually; only the margins need "eventually". This last step does not compare stochastic boundary distances with $1/N$: globality signs every exact move at once. Two families of moves are checked separately (H2). A move that empties its source is inadmissible by the in-bin convention, and could not improve anyway, since the result is a coarsening and DS11(b) refinement monotonicity is one-directional. A move into a destination with $\hat I_{\lambda\lambda}=0$ leaves the DS9-feasible class; it is inadmissible in-bin, and on $\Omega_4$ no such destination exists at all. Under the DS11 pseudo-inverse convention it would be evaluable and could strictly beat the optimum — see the boundary fixture below. ∎

Boundary, scope, and deployment consequence

The stronger finite assertion is false. On the support-minimal movable sample CE-DS-NONCENTERED-POPULATION-CUT-UNSTABLE-001, the raw $q^*$ labels have an exact improving gain $37/14608$. Thus $O(1/N)$-scale boundary effects are real; DS18.2 bypasses them through finite global selection rather than pretending they vanish pointwise.

The feasibility convention is load-bearing, which the audit serialized as CE-DS-NONCENTERED-SINGULAR-DESTINATION-001. On four exactly centered rows of this law's own support — $X=(-1,0,\tfrac12,\tfrac12)$, $Z=(-1,1,-\tfrac34,\tfrac14)$ — the exact global value over regular labelings is $1/12$, attained twice, and both attainers reach the single nuisance-singular labeling by one source-nonempty relocation whose DS11 pseudo-inverse value is $3/32$, an exact gain of $1/96$. Under a pseudo-inverse comparison domain no global regular optimum of that table is exchange-stable. DS18.2 is untouched — such tables are null under the law — but the statement must name the in-bin (DS9) convention, and any future ProfiledMarginPolicy surface must name it too. $N=4$ is minimal: at $K=3$, $N=3$ has three singleton cells and zero admissible relocations.

The theorem proves that the DS14 margin-certified branch is genuinely inhabited off (L), and gives a law-specific theorem-backed limiting companion rule. It does not prove that exchange ascent finds the basin, that every isolated DS17 root persists, that mix3 has a unique root, or that the result is robust to law/score estimation. No library compile surface follows before a fresh independent audit and a practical selection theorem.

Self-adversarial checks: population ties are null; empirical empty cells are excluded by feasibility and disappear under mass convergence; the limiting nuisance block is regular; bounded support supplies uniform integrability and (M4) with $\varphi(t)=\min(1,\sqrt{29}\,t/2)$ — audit-verified on 480 exactly computed slabs with worst measured ratio $1.9986$, and note the support's own diameter is $\sqrt{26.0345\ldots}$, not $\sqrt{26}$; duplicate full scores occur with probability zero; no first-order move approximation is used; exact scores and equal weights are load-bearing; no sample centering is performed, so every empirical second moment is about the origin. The exact root, exhaustive $N\le10$ falsification, unequal-weight/duplicate/tie/tiny-cell/singular controls, and the boundary fixture are reproduced by py/ds_noncentered_global_basin.py; the independent audit reproduces all of them from the law definition, plus the exact slab sweep, the codebook lattice scan, the $N\le11$ global optima and the finite-$N$ certificate, with py/audit_ds_noncentered_global_basin_transfer.py.

Stop-condition verdict. PROVED for an explicit off-(L) law, and independently audited. A regular DS17 root with fixed positive margins is the unique strict population global basin, and every sequence of finite global profiled optima transfers almost surely to exact exchange-stable empirical inhabitants of that basin, under the in-bin feasibility convention and at the computable rate above. The audit changes no deployment verdict: this remains a global-oracle existence theorem for one named law, and authorizes no src/ compile surface. The next dependency-blocking question is now sharp — is there a polynomial-time labeling rule whose output $\tilde z_N$ satisfies $\hat v_{3,N}-\hat\Phi_{D_s}(\tilde z_N)\to0$ a.s. on this law? Any such rule inherits the displayed finite-$N$ bound and hence the whole transfer.