A root equation gates regular self-consistency

Independent audit (31 Aug 2026, AUDITS/AUDIT-DS-STABLE-BASINS-001.md). All four nodes are hardened. The obstruction and eventual-emptiness theorem survive. The audit separates the root equation, which remains meaningful at a singular binned nuisance block, from regular tilt consistency $B_q^*=\beta$, which does not; restricts the LCM conclusions and Gaussian sign-split construction to their exact scopes; and demotes every finite root scan to windowed measured evidence. A numerical scan cannot decide the gate for a law, and a root is necessary, never sufficient, for empirical inhabitation.

Normalization (protocol A). Criterion: in-bin profiled $D_s$. Levels: population_quantizer (existence/non-existence of self-consistent rules) and empirical_to_population (the inhabitation consequence). Decision variables: ordinary one-point exchange-stable finite labelings (DS14's hypotheses — never stability-under-constraint) and population strip rules. Setting as in DS14/DS15: $d_\psi=d_\lambda=1$, $K\ge3$, $ES=0$, $E\|S\|^2<\infty$, $I=E[SS^\top]\succ0$, efficient score $\hat s=S_\psi-B^*S_\lambda$, $B^*=I_{\psi\lambda}/I_{\lambda\lambda}$. For $\beta\in\mathbb R$ write $T_\beta=S_\psi-\beta S_\lambda$; a strip rule at tilt $\beta$ is a $K$-cell interval partition of $T_\beta$ with positive masses. Neither (S) nor (R) is assumed anywhere in DS17.1–DS17.2; (L) is the DS15 conditional centering $E[S_\lambda\mid\hat s]=0$ a.s. Deployability question decided: is OPEN-DS-STABLE-BASINS(a) — almost-sure sequences of margin-certified exchange-stable labelings — inhabited on the DS15/DS16 class? Information-loss implication: on that class the certified branch retains nothing because it is empty; off the class the gate identity below is only a necessary population test. Conditional centering is a property of the population score law; it is not, and does not authorize, sample centering or recentering of score rows.

Lemma DS17.1a (tilt-residual identity; any law, any partition)

For any $K$-cell partition $q$ with masses $W_b>0$, centroids $(\mu_{\psi,b},\mu_{\lambda,b})$, and any $\beta$, set $t_b=\mu_{\psi,b}-\beta\mu_{\lambda,b}$. Then $\sum_bW_bt_b\mu_{\lambda,b}=I_{\psi\lambda}(q)-\beta I_{\lambda\lambda}(q)$, so whenever $I_{\lambda\lambda}(q)>0$,

$$B^*(I_q)-\beta=\frac{\sum_bW_bt_b\mu_{\lambda,b}}{I_{\lambda\lambda}(q)} .$$

For a strip rule at tilt $\beta$, $t_b=E[T_\beta\mid A_b]$ and the numerator is $E[h(T_\beta)S_\lambda]$, where $h=\sum_bt_b1_{A_b}$ is the non-decreasing step function sending each $T_\beta$-interval to its conditional mean (strictly increasing values across cells, for an atomless law with positive masses). Proof: expand the definitions; for the strip form use the tower property $\sum_bW_bt_b\mu_{\lambda,b} =E[\,E[T\mid\sigma(q)]\,E[S_\lambda\mid\sigma(q)]\,]=E[h(T)S_\lambda]$, since $h(T)=E[T\mid\sigma(q)]$ is $\sigma(q)$-measurable and bounded ($|h|\le\max_b|t_b|<\infty$). ∎

Two immediate readings. (i) Call the numerator condition $E[h(T_\beta)S_\lambda]=0$ root consistency. When $I_{q,\lambda\lambda}>0$, it is equivalent to the regular notion of tilt consistency $B^*(I_q)=\beta$. When $I_{q,\lambda\lambda}=0$, $B^*(I_q)$ is undefined and only the root equation may be stated. A self-consistent efficient-Voronoi rule with a positive nuisance block is regularly tilt-consistent at $\beta=B^*_q$ by construction. (ii) At any regular tilt-consistent pair the projected centroids satisfy $e_b=\mu_{\psi,b}-\beta\mu_{\lambda,b}=t_b$ identically — projection linearity, no law hypothesis — so the nearest-centroid cuts are exactly the Lloyd midpoints of $\operatorname{law}(T_\beta)$: self-consistency decomposes into "cuts are Lloyd-stationary for $T_\beta$" plus the scalar root equation $E[h(T_\beta)S_\lambda]=0$. This is the gate every later scan tests.

Theorem DS17.1 (conditional-centering obstruction; population)

Let $P$ be atomless, in class (L), with $ES=0$, $E\|S\|^2<\infty$, $I\succ0$. Then every root-consistent strip rule has $I_{q,\lambda\lambda}=0$, at every tilt $\beta$ and every $K\ge2$. Equivalently, no regular tilt-consistent strip rule exists. In particular, there is no full-rank bounded-study stationary rule with pairwise-distinct projected centroids. No LCM, ellipticity, independence, or log-concavity is assumed: (L) alone.

Proof. Write $\delta=\beta-B^*$, so $T_\beta=\hat s-\delta S_\lambda$.

Case $\delta=0$. Cells are $\hat s$-measurable, so $W_b\mu_{\lambda,b}=E[S_\lambda1_{A_b}]=E[\,E[S_\lambda\mid\hat s]1_{A_b}]=0$ by (L): $I_{\lambda\lambda}(q)=\sum_bW_b\mu_{\lambda,b}^2=0$ directly.

Case $\delta>0$. Condition on $\hat s$: the map $x\mapsto h(\hat s-\delta x)$ is non-increasing ($h$ non-decreasing). The Chebyshev association inequality for a non-increasing transform — for an i.i.d. conditional copy $X'$ of $X=S_\lambda$ given $\hat s$, $2\,\mathrm{Cov}(h(\hat s-\delta X),X\mid\hat s) =E[(h(\hat s-\delta X)-h(\hat s-\delta X'))(X-X')\mid\hat s]\le0$ termwise — gives, with (L) killing the product of conditional means,

$$E[h(T_\beta)S_\lambda\mid\hat s] \le E[h(T_\beta)\mid\hat s]\,E[S_\lambda\mid\hat s]=0 \quad\text{a.s.}$$

Integrating, $E[h(T_\beta)S_\lambda]\le0$. Now suppose $I_{\lambda\lambda}(q)>0$ and root consistency. By DS17.1a the numerator is exactly $0$, so the conditional inequality is an equality a.s. Equality in the termwise-nonpositive covariance forces, for $P_{\hat s}$-a.e. value of $\hat s$: $(h(\hat s-\delta X)-h(\hat s-\delta X'))(X-X')=0$ a.s., i.e. $h(\hat s-\delta S_\lambda)$ is a.s. constant given $\hat s$ (if $S_\lambda$ is conditionally degenerate the same constancy holds trivially). Since $h$ takes distinct values on distinct cells, every cell indicator $1_{A_b}(T_\beta)$ is then a.s. $\hat s$-measurable, and the Case-$\delta=0$ computation applies verbatim: $\mu_{\lambda,b}=0$ for all $b$, so $I_{\lambda\lambda}(q)=0$ — contradiction.

Case $\delta<0$. $x\mapsto h(\hat s-\delta x)$ is non-decreasing; the inequality reverses ($E[h(T)S_\lambda]\ge0$) and the same equality analysis applies. Finally, if $I_{\lambda\lambda}(q)=0$, the asserted conclusion already holds (and PSD also forces $I_{\psi\lambda}(q)=0$); no quotient or value of $B_q^*$ is used in this singular case. ∎

Consequences. Under (L) a bounded-study stationary rule with $I_q\succ0$ (full rank) cannot have pairwise-distinct projected centroids: distinctness plus tie-nullity (below) would make it a genuine tilt-consistent strip rule, forcing $I_{\lambda\lambda}=0$ and, by $I_{\psi\lambda}^2\le I_{\psi\psi}I_{\lambda\lambda}$ (PSD), rank $\le1$. Margin-compatible stationary configurations on class (L) are therefore all wasted-cell structures in the DS12 sense — the merged branch of DS17.3.

Lemma DS17.0 (pathwise DS14′)

Fix $K$ and rational $\kappa,c_0,\gamma>0$, and let $P$ satisfy (M1) and (M4). There is one probability-one event $\Omega_0$ — the intersection, over the countable family of rational constants and integer truncation radii, of: the SLLN events for $P_N\|S\|^2$ and $P_NS$; the DS14 Step-2 Glivenko–Cantelli event over the fixed slab family; and the C1 uniform moment law over the compact affine-max class with parameters bounded by $(c_0,\kappa,\gamma,M)$ — such that on $\Omega_0$ the following holds for every (arbitrarily data-dependent, merely $\omega$-wise) choice of one-point exchange-stable $K$-cell labelings $z^{(N)}$, $N\in\mathcal N$, $|\mathcal N|=\infty$, whose margins (M2)+(M3)+(M5) hold at $(c_0,\kappa,\gamma)$ for every $N\in\mathcal N$: along a sub-subsequence of $\mathcal N$ the companion parameters converge, and the limit $q^*$ is a self-consistent efficient-Voronoi quantizer with $\lambda_{\min}(I_{q^*})\ge\kappa$, cell masses $\ge c_0$, and pairwise-distinct projected centroids separated by $\ge\gamma$.

Proof. DS14's five steps consume only: the margins at the indices actually used; the uniform laws — all over fixed function classes after the audit's repair (AUDITS/AUDIT-DS-POPULATION-BRIDGE-001.md §8), hence valid simultaneously for every candidate sequence on $\Omega_0$; and compactness of the explicit parameter set, which supplies the convergent sub-subsequence. Restricting the index set to $\mathcal N$ leaves every step intact (the sample is the same; limits along $\mathcal N$ use the same events). No step selects a null set depending on the sequence, so the conclusion is pathwise on $\Omega_0$; measurable selection is not needed because the argument runs at fixed $\omega$. $\lambda_{\min}$ passes to the limit by its 1-Lipschitz continuity in operator norm; separation and masses pass by DS14 Step 3. This is the DS16 §"proof of (1)" quantifier idiom applied to DS14. ∎

Theorem DS17.2 (inhabitation disproved on the whole class; eventual emptiness)

Let $P$ be in class (L) with (M1) and (M4) and $I\succ0$ — the DS15/DS16 law class satisfies this, and neither (S) nor (R) is needed. Then almost surely, for every rational $\kappa,c_0,\gamma>0$ there is $N_0(\omega)<\infty$ such that for all $N\ge N_0$ no one-point exchange-stable $K$-cell labeling of the sample satisfies (M2)+(M3)+(M5) at $(c_0,\kappa,\gamma)$ at all. In particular OPEN-DS-STABLE-BASINS(a) is disproved: on the entire conditionally centered class there is no almost-sure margin-certified exchange-stable sequence, for any margins; DS14's hypothesis set is asymptotically empty, and real margins at real constants are covered by shrinking them to rationals.

Proof. Work on $\Omega_0$. If some $(c_0,\kappa,\gamma)$-margin- compatible stable labeling existed for infinitely many $N$, choose one per such $N$ ($\omega$-wise) and apply DS17.0: a limit $q^*$ exists — bounded-study stationary with $\lambda_{\min}(I_{q^*})\ge\kappa>0$ and pairwise-distinct projected centroids. By DS14 Step 4 the limit is the nearest-projected-centroid rule of its own centroids and metric (the audited identification — not bare DS12, whose stationarity alone would tolerate coincident centroids); for $d_\psi=1$ the scalar metric $S_\psi(I_{q^*})^{-1}>0$ cancels, so $q^*$ is a strip rule at tilt $\beta^*=B^*_{q^*}$, genuinely tilt-consistent, with tie sets $P$-null by (M4). Theorem DS17.1 gives $I_{q^*,\lambda\lambda}=0<\kappa\le\lambda_{\min}(I_{q^*})\le I_{q^*,\lambda\lambda}$ — contradiction. Union over the countable rational constants. ∎

Corollaries (instances). Jointly Gaussian and atomless elliptical laws with $\Sigma\succ0$ are in (L) (linear conditional means) and satisfy (M4) (below), so the canonical law is covered. Product laws $S_\psi\perp S_\lambda$, $ES_\lambda=0$, are in (L) via independence ($B^*=0$, $\hat s=S_\psi$) — every bimodal-nuisance product law scanned below is an instance. For centered LCM laws with nonsingular covariance (linear conditional means along every nondegenerate tilt, e.g. elliptical), (L) follows by applying LCM to $\hat s$, since $\operatorname{Cov}(S_\lambda,\hat s)=0$. There is also an independent one-line second proof: cell means of a strip rule are collinear, $\mu_b=\theta\,t_b$ with $\theta=\mathrm{Cov}(S,T_\beta)/\mathrm{Var}(T_\beta)$ and $\mathrm{Var}(T_\beta)\ge\lambda_{\min}(I)(1+\beta^2)>0$, so $I_q=(\sum_bW_bt_b^2)\,\theta\theta^\top$ has rank $\le1$ and $\lambda_{\min}(I_q)=0$; moreover the tilt-consistency equation on a Gaussian is the Möbius fixed point $\beta=(a-\beta c)/(c-\beta d)$, i.e. $\beta^2d-2\beta c+a=0\Longleftrightarrow\mathrm{Var}(T_\beta)=0$, with discriminant $4(c^2-ad)<0$: no real solution at any correlation.

(M4) for atomless elliptical laws ($d=2$, $\Sigma\succ0$). Writing $S=R\Lambda U$ with $U$ uniform on the circle: for unit $v$, $P(|v^\top S-c|\le t)\le P(R\le\sqrt t\,k_1)+\sup_{r>\sqrt t k_1} P_U(|v^\top\Lambda U-c/r|\le t/r)\le P(R\le\sqrt tk_1)+k_2t^{1/4}$ — the angular arc a slab cuts from an ellipse of radius $r$ has measure $O(\sqrt{t/r})$ even at tangency — and $P(R\le\varepsilon)\downarrow0$ by atomlessness at the origin: a uniform $\varphi(t)\downarrow0$.

Theorem DS17.3 (LCM merged-branch classification: margins survive only in wasted cells, and are never compilable)

Let $P$ be atomless with (M4)-tie-nullity and LCM on the relevant tilt range, and let $q$ be bounded-study stationary with $W_b>0$ and $I_q\succ0$ (the merged branch: (M5) dropped). Then:

1. the projected centroids are not pairwise distinct (else DS12 deployability (i)+(ii) reproduces $q$ as a strip rule and the LCM rank bound — or Theorem DS17.1, since LCM$\Rightarrow$(L) — contradicts $I_q\succ0$);
2. the reduced rule $\rho'$ (merge coincident-$e$ groups; DS12) is a genuine $T_{B^*_q}$-interval rule with $K'\le K-1$ cells and $\operatorname{rank}(I_{\rho'})\le1$, so $\lambda_{\min}(I_{\rho'})=0$: the compilable reduction never carries a nuisance margin;
3. value identity: $\Phi(q)=\sum_{\text{groups }g}W_gE[e_q(S)\mid G_g]^2$ with $e_q(s)=s_\psi-B^*_qs_\lambda$ — the between-value of $\operatorname{law}(e_q(S))$ on at most $K-1$ intervals (proof: $\Phi(q)=\min_B\sum_bW_b(\mu_{\psi,b}-B\mu_{\lambda,b})^2$ attained at $B=B^*_q$ by the normal equation, $e_b$ constant on groups, and $\sum_{b\in g}W_be_b=E[e_q1_{G_g}]$); always $\Phi(q)\le v_K$ (DS15 Lemma 1);
4. nonemptiness: on $N(0,I_2)$ the sign-split family — the threshold cell $\{S_\psi\ge0\}$ plus any nontrivial measurable split of the left half determined by $S_\lambda$ (e.g. $\{S_\lambda>v\}$ versus $\{S_\lambda\le v\}$) — is bounded-study stationary with $B^*_q=0$, coincident left projected centroids, $I_q=\operatorname{diag}(2/\pi,\,I_{\lambda\lambda}(\text{split}))\succ0$, $\lambda_{\min}$ up to $1/\pi$ (at $v=0$), and profiled value exactly $v_2=2/\pi$ for every member. The population constraint class $\{\lambda_{\min}(I_q)\ge\kappa\}$ of DS16's $v^*(\kappa)$ is therefore nonempty for $\kappa\le1/\pi$ on the canonical law — the audit's attainment subproblem has a nonvacuous feasible set. This proves neither attainment/continuity of $v^*(\kappa)$ nor existence of empirical stable sequences. For this explicit family, the loss against the scalar Gaussian three-bin optimum is the measured quantity $v_3-v_2\approx0.1732$ ($=0.809826-2/\pi$); it is not a lower bound for every stationary inhabitant.

Exact-rational verification: the $K=3$ sign-split sibling of CE-DS-POP-WASTED-CELLS-001 on the same 8-atom law has $I_q=\operatorname{diag}(4,9/8)$, $\Phi=4$ equal to the $K'=2$ group between-value, projected centroids $(-2,-2,2)$, zero first-order violations, and singular reduced nuisance block — serialized as CE-DS-LCM-SIGNSPLIT-MARGIN-001 and CI-pinned. Under bare (L), conclusion 1 still holds (via Theorem DS17.1), while conclusion 2's rank collapse of the reduced rule is LCM-scoped: a non-LCM (L)-law may leave its reduced strips with a nonzero nuisance block, and only the fine configuration's coincidence is forced.

The audit also records the support-minimal atomic boundary at $N=K=3$ as CE-DS-LCM-SIGNSPLIT-MINIMAL-001. It has the same coincident-centroid and singular-reduction mechanism, but bounded-study stationarity is vacuous on its singleton atoms. The eight-atom fixture remains the structured symmetric exact control; it is not globally support-minimal.

Corollary DS17.4 (the fixed-point gate, and what it costs off the class)

For any atomless law with (M4): if OPEN-DS-STABLE-BASINS(a) full-triple inhabitation holds at $(\kappa,c_0,\gamma)$, then there exist $|\beta|\le2M/\kappa$ and a Lloyd-stationary $K$-interval quantizer of $\operatorname{law}(T_\beta)$ (any centroid-midpoint fixed point, not only the optimum) whose strip rule $q$ satisfies the root equation $E[h(T_\beta)S_\lambda]=0$ with $\lambda_{\min}(I_q)\ge\kappa$, masses $\ge c_0$, and $t$-mean separation $\ge\gamma$. DS17.1 proves the gate empty on all of class (L). Numerical root searches can probe but cannot decide this gate for a law. Off the class, the mix3 scan finds a root at $\beta=0$, cuts $\pm1.00476$, with measured $I_{\lambda\lambda}=\lambda_{\min}=1.7364$ and value equal to the efficient interval optimum to the reported tolerance. This is per-law evidence that a margin may have negligible price; it is not a general theorem and does not establish the empirical transfer required by OPEN-DS-MARGINS-NONCENTERED(a).

Self-adversarial notes (protocol G)

• Strictness and ties. Finite labelings may carry exact ties and duplicates; nothing in DS17.2 constrains them — stability is a hypothesis. Population tie sets are (M4)-null; the equality analysis in DS17.1 handles conditionally degenerate $S_\lambda$ explicitly.
• Singleton/empty cells. Excluded at the population limit by (M2); finite singletons are allowed and irrelevant.
• Duplicate scores. Allowed; DS14's Steps 1–3 already cover split atoms.
• Singular information. $I\succ0$ is a hypothesis on the law; post-move singular states never enter (DS13 covers them upstream in DS14).
• Nuisance singularity. The conclusion is nuisance singularity; the margin hypothesis (M3) uses $\lambda_{\min}\le I_{\lambda\lambda}$, so the contradiction is against the weaker block bound — no gap.
• Atomic laws. DS17.1 needs atomlessness through the strict increase of interval means and DS12/DS14's own hypotheses; the 8-atom fixture verifies algebra, never the population statement (DS12 necessity is vacuous on atoms — wasted-cells precedent). The audit's support-minimal three-atom fixture makes that boundary explicit.
• Hidden compactness. Only DS14's audited explicit compact class is used.
• First-order-to-finite jumps. None: the finite half is DS14 verbatim; DS17 adds population algebra plus the pathwise quantifier.
• Empirical-to-population jumps. Only DS17.0's fixed-class events — the audit-repaired uniform laws; no new uniform law is introduced.
• Score-estimation error. Excluded; exact scores (“Estimated scores and classifiers” unchanged).
• New-event extension. DS17 strengthens the refusal: on class (L) there is asymptotically nothing to compile in the certified branch; no new inductive claim is made.
• Why the census witnesses don't contradict DS17.2. The margin-retaining stable states at $N\le14$ (CE-DS-STABLE-MARGIN-RETAINING-001) are pre-asymptotic: DS17.2's $N_0(\omega)$ is finite but not uniform, and the grid laws are atomic emulations. The theorem predicts their margins die as $N$ grows — exactly the DS16 library measurement (funnel at $N=100$–$1000$) and the geometry scan below.

Measured (gate scans; falsification evidence that failed to falsify)

Instrument py/ds_stable_basins.py (closed-form Gaussian-mixture strip moments, validated in selftest against the public scorequant.IntegrationSource tensor Gauss–Legendre quadrature at $10^{-9}$ and the exact 8-atom rationals; artifacts with full provenance under artifacts/DS-STABLE-BASINS/):

• Gaussian laws ($\rho\in\{0,0.6,0.9\}$ and an anisotropic case): at every tilt on a 241-point grid, $\lambda_{\min}(I_q(\beta))\le2.4\times10^{-16}$ (exact rank one); the tilt-residual is zero-free (discriminant $<0$); the Möbius iteration never converges (elliptic orbit) — row N-DS-BASINS-GAUSS.
• Product bimodal (L)-laws ($m\in\{0.75,1,1.5,2,3\}$, $s=0.4$): the branch-tracked root scan over $\beta\in[-2.5,2.5]$, tracking up to 3 Lloyd-stationary branches per tilt (asymmetric mode-splitting branches included), found zero self-consistent roots in that finite window and branch set. The window is not the full compact gate range for an unstated $\kappa$, and the branch cap cannot exclude isolated roots.
• Dependent (L)-laws xcorr(c) (equal mixtures of $\pm c$-correlated standard Gaussians, $c\in\{0.5,0.8,0.95\}$; (L) by branch symmetry, LCM fails off $\beta=0$): zero roots — both scans in row N-DS-BASINS-CLASS-L-ROOTS. Eight (L)-laws, three structural families, zero gate inhabitants: the theorem's prediction, tested before its proof was trusted.
• Sign-split family on $N(0,I_2)$: $\Phi-2/\pi=0$ to machine precision across the split parameter $v\in[-1,1]$, $I_{\psi\lambda}=0$, $\lambda_{\min}$ maximal $=1/\pi$ at $v=0$; $v_3-v_2=0.173206$ — row N-DS-BASINS-SIGNSPLIT.
• mix3 (off-class control): one root found — $\beta=0$, cuts $\pm1.00476$, $\lambda_{\min}=1.7364$, value $2.68936=v_3$ to $10^{-6}$ (price $\approx0$), zero Monte-Carlo rule violations; the finite-difference family Hessian at the root has gradient $0$ and eigenvalues $(-2.146,-0.258,-0.186)$: a strict local maximum of the population value in the rule family, the second-order input any off-class transfer theorem (OPEN-DS-MARGINS-NONCENTERED(a)) would consume — row N-DS-BASINS-MIX3.
• Seeded library ascent at scale (public API, $N\in\{300,1000,3000\}$, 3 reps): all 63 terminals exchange-stable. gauss06: $\lambda_{\min}\in[5\times10^{-5},0.012]$ with $N\hat I_{11}\approx0.3$–$7$ at every seed — the disproof's finite face. mix3: the population-fixed-point seeding terminates at the same terminal as the efficient and k-means seeds in 9/9 runs, with $\lambda_{\min}=1.73$–$1.83$ and log-gap $\le10^{-5}$ at $N\ge1000$; one random-seed run stranded at a degenerate inferior terminal ($\lambda_{\min}=5\times10^{-4}$, gap $0.21$) — off-class margins are free at the good optimum, but bad local terminals still exist — row N-DS-BASINS-LIBRARY.
• Geometry of the DS16 terminals (420 recorded ascent terminals plus the margin-retaining witness): the witness agrees exactly with its own companion rule at $N=8$ ($\hat B^*=0.497$, separation $0.325$); on centered06 only $2\%$ of stable terminals are companion-exact (median disagreement $18\%$, median $\lambda_{\min}=0.034$), on mix3 $61\%$ are exact (median $\lambda_{\min}=1.06$) — the class boundary of DS17.1 visible in the finite states — row N-DS-BASINS-GEOMETRY.

Verdict: OPEN-DS-STABLE-BASINS(a) and the compile consequence

• Disproved, on the whole class. On every (L)+(M1)+(M4) law — all of DS15/DS16's class, jointly Gaussian and elliptical laws in particular — the margin-certified branch is almost surely eventually empty: not merely uninhabited by sequences, but devoid of any single margin-compatible exchange-stable labeling for all large $N$. The DS14 companion path on this class is vacuous asymptotically; compile_quantizer's refusal needs no certificate carve-out there, and a margin-constrained solver (an earlier, since-closed question/OPEN-DS-STABLE-BASINS(c)) cannot terminate at ordinary-stable margin states eventually — any terminal it certifies is stable only under the constraint.
• The merged branch survives but compiles to nothing. Dropping (M5), margin-compatible stationary configurations exist on the canonical law (sign-split family, $\lambda_{\min}$ up to $1/\pi$), all wasted-cell structures whose compilable reduction has $\lambda_{\min}=0$ and whose value is pinned at $v_2$ in every scanned instance — retention is bought only by spending cells on profiled-information-free splits.
• Off the class the population gate can be inhabited and may be cheap. mix3's gate inhabitant is the efficient optimum itself ($\lambda_{\min}=1.74$, price $\approx0$) in the measured family. This one law does not support a general claim that certification is free. The deployment-relevant remainder moves to OPEN-DS-MARGINS-NONCENTERED branch (a) (non-centered laws) with the DS17.4 gate identity as its population test, and to the DS16 attainment question, whose feasible class DS17.3(4) proves nonempty.
• Deployment statement (required by the study). An inhabited DS14 sequence: impossible on class (L) (this theorem). A finite diagnostic: still legitimate as a diagnostic (the $N=8$ witness is even companion-exact) but now known to be transient on the class — it certifies nothing asymptotic. A constrained terminal: the only object a margin-constrained solver can deliver on the class, and it is not ordinary-exchange-stable eventually; any future ProfiledMarginPolicy surface must present it as constrained, priced ($\hat v_K-\hat\Phi_s$), and non-inductive on this class.

Stop condition 2 (disproved) of the study is hit on the study's own class, with the obstruction's boundary serialized (CE-DS-LCM-SIGNSPLIT-MARGIN-001: (M5) is load-bearing) and the off-class escape measured (mix3 root). The isolated remainder is OPEN-DS-MARGINS-NONCENTERED(a)'s non-centered inhabitation/transfer, gated by DS17.4, plus DS16's constrained-value attainment — recorded in the OPEN-DS-STABLE-BASINS rescope.