Regularity margins have an information price

Audit (30 Aug 2026, AUDITS/AUDIT-DS-STABLE-MARGINS-COMPILE-001.md): all three claims are hardened. The PRICE/FUNNEL/FLOOR theorem is verified after replacing a pointwise SLLN at a data-dependent centroid limit by a uniform strong law on compact tilt--codebook sets, making the all-labelings event explicit, and fixing the raw-label/exact-centering convention in the FLOOR. The compile claim now says only currently established in the registered theory, not mathematical uniqueness: DS14 is a theorem for certified sequences, and a single finite certificate supplies diagnostics but no population guarantee. The measured census range is corrected from 18--944 to 5--944, and its reported library gap interval is identified as an aggregate summary rather than a run-wise bound.

Setting: exactly DS15's — $S=(S_\psi,S_\lambda)\sim P$ on $\mathbb R^2$, $ES=0$, $E\|S\|^2<\infty$, $I\succ0$, class (L) (conditional centering) + (S) (scalar regularity of $\operatorname{law}(\hat s)$); (R) is not needed here except where DS15's conclusion (1) is imported for a comparison. Equal weights, centered score rows $\tilde S_i=S_i-\bar S_N$ in every empirical information calculation, $K\ge3$ ($=d_\psi+d_\lambda+1$; see the cardinality restatement below), feasible labelings = nonsingular binned nuisance block $\hat I_{\lambda\lambda}>0$. $v_K$, $J^*$, $w^*_b$, $\hat v_K$, $\mathrm{btw}(\hat s_N;z)$ as in DS15.

Cardinality, restated (library commit 891bbf3, 30 Aug 2026). The registered condition $K\ge d_\lambda+2$ is a $d_\psi=1$ coincidence. The mechanism is $\sum_b m_b=\hat\mu$ (with $m_b=\sum_{i\in b}w_is_i$): the binned moment vectors of an exactly centered sample satisfy one linear relation, so $\operatorname{rank}(I_z)\le K-1$, and a nonvacuous profiled value needs $K-1\ge d=d_\psi+d_\lambda$, i.e. $K\ge d_\psi+d_\lambda+1$ on a centered sample. At $d_\psi=1$ this is exactly $K\ge d_\lambda+2$, which is why CE-DS-MARGINS-RANK-VACUITY-001 (a $d_\psi=1$ witness) suggested the narrower formula; its consequences[0] line carries that $d_\psi=1$ restriction implicitly.

The algebraic rank bound can be one larger for a matrix built from rows with nonzero empirical mean. That is not an alternative Fisher-score regime: scores have mean zero, and ScoreQuant's empirical information uses exact centering. No off-centered compile claim is inferred from that matrix fact.

The question

DS15 shows free global finite $D_s$ optima converge to the nuisance-degenerate efficient-score interval quantizer: (M2) holds, (M3) fails. The library's optimizer returns one-point exchange-stable, generally non-global states (DS-EXCHANGE-TERMINATES), and DS15 asserts nothing about them. OPEN-DS-MARGINS-NONCENTERED's deployment half asks: do those states retain the DS14 margins, at what information cost $v_K-\hat\Phi_s$, and does a theorem-backed inductive compile rule for profiled criteria exist beyond the projected efficient-score interval rule?

The first observation is that the question is not a universal quantification: global optima are themselves exchange-stable, so DS15 already exhibits stable sequences along which (M3) fails. What decides the compile question is (i) whether the margins are priced — whether any labeling carrying them must sit a definite distance below the supremum — and (ii) which regime the optimizer's terminal states actually occupy.

Theorem (margin price and the value funnel)

Under (L)+(S), $d_\psi=d_\lambda=1$, $K\ge3$, equal weights, almost surely:

1. (Price.) For every $\kappa>0$ there is $\delta(\kappa)>0$, depending only on $(P,K,\kappa)$, with $$\limsup_N\; \sup\bigl\{\hat\Phi_s(z)\;:\;z\ \text{feasible},\ \hat I_{\lambda\lambda}(z)\ge\kappa\bigr\} \;\le\; \limsup_N\; \sup\bigl\{\mathrm{btw}(\hat s_N;z)\;:\; \hat I_{\lambda\lambda}(z)\ge\kappa\bigr\} \;\le\; v_K-\delta(\kappa).$$ Since $\lambda_{\min}(\hat I_N)\le\hat I_{\lambda\lambda}$, the same cap holds under the DS14 conditioning margin (M3) at level $\kappa$. The hypothesis is a margin, not stability or optimality: every margin-carrying labeling — exchange-stable or not, optimizer-produced or not — pays at least $\delta(\kappa)$ of profiled information relative to the unrestricted supremum $v_K$ that free global optima attain under (L)+(S)+(R) (DS15 conclusion 1).
2. (Funnel.) Any sequence of feasible labelings with $\hat\Phi_s(z^{(N)})\to v_K$ — in particular any asymptotically value-optimal solver output, from any seed, stable or not, global or not — satisfies: cells converge in sample measure to $J^*$, min cell mass $\to\min_b w^*_b>0$ ((M2) holds; singletons die out), $\hat I_{\lambda\lambda},\hat I_{\psi\lambda}\to0$, hence $\lambda_{\min}(\hat I_N)\to0$ ((M3) fails), and the (M5) object $\hat B_z=\hat I_{\psi\lambda}/\hat I_{\lambda\lambda}$ rides the same $0/0$ as in DS15. DS15's conclusions (2)–(3) are therefore value-topological, not properties of exact global optimality.
3. (Floor.) For every fixed measurable $K$-cell quantizer $q$ on the raw score $S$, with $W_b>0$ and $I_{q,\lambda\lambda}>\kappa$, label row $i$ by $q(S_i)$ and compute its empirical information from the centered rows $\tilde S_i$. Those labelings are eventually feasible with $\hat I_{\lambda\lambda}\ge\kappa$ and $\hat\Phi_s\to\Phi(q)$. Hence the supremum in (1) is asymptotically at least $v^{*+}(\kappa):=\sup\{\Phi(q):I_{q,\lambda\lambda}>\kappa\}$, and $v^{*+}(\kappa)\le v_K-\delta(\kappa)$; DS15's margin-compatible optimum $v^*(\kappa)$ (over $\lambda_{\min}(I_q)\ge\kappa$, a subclass) obeys $v^*(\kappa)\le v^{*+}(\kappa)$. This is a supremal lower bound: it asserts neither attainment nor one-sided continuity of either constrained value in $\kappa$.

Proof

Throughout, work on one probability-one event where: $\hat v_K(\hat s_N)\to v_K$ and the tilt-Lipschitz bound of DS15 Proposition 5 holds (its proof needs only (M1)-type moments and (S); the same argument gives $\hat v_{K-1}\to v_{K-1}$); $\hat B^*_N\to B^*$ and the SLLN holds for $\|S\|^2$, $S_\lambda^2$, and the fixed cells $1_{J^*_b}$ weighted by $S_\lambda$; the Glivenko–Cantelli law holds over the fixed VC class of tilted half-planes $\{s:s_\psi-\beta s_\lambda\le c\}$, $(\beta,c)$ in a compact neighborhood of $(B^*,\cdot)$, both unweighted and with integrable envelope $|S_\lambda|$. For the signed weight, apply the VC theorem to $S_\lambda^+1_H$ and $S_\lambda^-1_H$ separately; compact tilt control is already included because all $(\beta,c)$ half-planes form one fixed finite-VC class, and $E|S_\lambda|<\infty$ follows from the second moment. Also include the uniform strong law

$$\sup_{\beta\in\mathcal B,\ C\in[-R,R]^K} \left|(P_N-P)\min_{c\in C}(S_\psi-\beta S_\lambda-c)^2\right|\to0$$

for every compact $\mathcal B$ containing $B^*$ and finite $R$. This follows by truncation and finite parameter nets: the class is continuous in the finite-dimensional parameter $(\beta,C)$, and on each compact set has an integrable envelope $A_R(1+\|S\|^2)$. Finally include the empirical DP boundaries and masses of $\hat J^*_N$ converge to $J^*$'s (the audit-hardened boundary/mass consistency ingredient of DS15 Proposition 6, which needs only (S)).

Lemma DS16.1 (empirical grouping rigidity, uniform). For every $\varepsilon>0$ there are $\delta_0>0$ and an a.s. finite $N_0$ such that for all $N\ge N_0$ and every $K$-grouping $z$ of the sample indices with $\mathrm{btw}(\hat s_N;z)\ge \hat v_K-\delta_0$: after relabeling, $\hat P_N(z\text{-cell}_b\,\Delta\,\{\hat s_N\in J^*_b\})\le\varepsilon$ for every $b$, and $\hat W_b(z)\ge w^*_b-2\varepsilon$.

Proof. This is DS15 Lemma 3's five-step architecture run on the empirical scalar sample, with groupings in place of measurable partitions — the one genuine gap between the two settings, closed as follows. Write $x_i=\hat s_{N,i}$, $\hat P_N\hat s^2=\mathrm{btw}(z)+\mathrm{WSSE}(z)$, so the hypothesis reads $\mathrm{WSSE}(z)\le\hat W_K+\delta_0$ with $\hat W_K=\hat P_N\hat s^2-\hat v_K$ the optimal empirical $K$-grouping SSE (attained by sorted intervals — the 1-D contiguity fact of DS15 Proposition 4, verified on the full lattice in the identity suites).

(i) Inactive-cell budget. For $\eta>0$ call $b$ active when $\hat W_b\ge\eta$. Uniform integrability transfers: for any $T$, $\sup_{\hat P_N(B)\le m}\hat P_N[\hat s^21_B]\le 2\bigl(\hat P_N[s_\psi^21_{s_\psi^2>T}]+\hat B_N^{*2}\hat P_N[s_\lambda^21_{s_\lambda^2>T}]\bigr)+2(1+B^{*2})Tm+o(1)$, and the tail terms converge to their population values by the SLLN on fixed functions; so inactive cells contribute $\tau(K\eta)+o(1)$ to the between value with $\tau(m)\downarrow0$.

(ii) Deletion comparison and (iii) full support. If one cell has mass below $\eta$, merge it into a bounded active cell. The increment in SSE is bounded by the inactive cell's second-moment contribution plus its mass times an active-centroid bound, hence by $\rho(\eta)+o(1)$, where $\rho(\eta)\downarrow0$ by (i). The resulting grouping uses at most $K-1$ cells, so its SSE is at least $\hat W_{K-1}$. Therefore $\hat W_{K-1}-\hat W_K\le\delta_0+\rho(\eta)+o(1)$. But $\hat W_{K-1}-\hat W_K\to W_{K-1}-W_K>0$ (value convergence at both budgets; strict positivity from (S), whose positive density near the optimal boundaries in particular rules out a finite-support collapse). Choose $\eta$, then $\delta_0$, below this gap: every cell is active. This also closes the apparent $K$-versus-$K-1$ loophole; an empty cell is simply the $\eta=0$ case.

(iv) Centroid convergence, uniform over groupings. This is the standard almost-minimizer rigidity principle for k-means codebooks (Rakhlin--Caponnetto 2006 supplies direct bounded-source prior art), with the grouping-to-nearest-center reduction made explicit here. Suppose groupings $z^{(N_j)}$ with $\mathrm{WSSE}\le\hat W_K+\delta_j$, $\delta_j\to0$, had active-centroid sets escaping a neighborhood of $C^*$. Active centroids are bounded (moment bound), so a subsequential limit set $C'\ne C^*$ exists. The nearest-point SSE is grouping-independent: $\mathrm{SSE}_{\hat P_N}(c(z))\le\mathrm{WSSE}(z)$: reassigning every observation to a nearest centroid cannot increase SSE, even when duplicate atoms were split among cells. By the compact-set uniform law displayed above, simultaneously for the data-dependent $\hat B_N^*$ and all bounded codebooks, $\mathrm{SSE}_{P}(c(z^{(N_j)})) \le \hat W_K+\delta_j+o(1).$ Continuity in the codebook then gives $\mathrm{SSE}_P(C')\le W_K$, contradicting (S)-uniqueness when $C'\ne C^*$. This uniform law is essential: a pointwise SLLN for the random subsequential limit $C'$ would be invalid. Thus near-optimal groupings have centroid sets uniformly convergent to $C^*$.

(v) Misassignment slabs. For a grouping $z$ with centroids $\gamma$-close to $C^*$: a point on cell $a$'s side at distance $\ge t$ from the midpoint of $(c_a,c_b)$ but assigned to $b$ pays excess $(x-c_b)^2-(x-c_a)^2\ge 2t\,g/2$ with $g=$ the minimal $C^*$ centroid gap, once $\gamma<g/4$; so the $\hat P_N$-mass misassigned beyond the $t$-slabs is $\le(\delta_0+o(1))/(tg/2)$, while the slabs themselves lie inside fixed $(t+\gamma)$-slabs around the $C^*$ midpoints whose empirical mass converges to the population slab mass, small by (S)-atomlessness. Choosing $t$ then $\delta_0$ gives the symmetric difference bound against the empirical Voronoi cells of $C^*$; replacing them by $\hat J^*_N$'s cells (boundary/mass consistency) and by the fixed sets $\{\hat s_N\in J^*_b\}$ costs empirical slab masses that vanish by the same GC-plus-atomlessness argument. Masses follow: $\hat W_b\ge \hat P_N(\hat s_N\in J^*_b)-\varepsilon\to w^*_b-\varepsilon$. ∎

Proof of (1). Define a supremum over an empty feasible set as $-\infty$. For each $N$, both suprema are maxima of finitely many extended-real measurable functions of the sample, so they are measurable. Intersecting the probability-one uniform-law events above over integer compact radii and rational tolerances produces one measurable event on which the argument is pathwise and simultaneous over all labelings at every $N$. Consequently the conclusion holds for every labeling sequence selected on that event, including a data-dependent selection; no separate exceptional set is chosen per sequence. When the constrained set is nonempty, let $z^{(N)}$ attain the $\mathrm{btw}$-maximum subject to $\hat I_{\lambda\lambda}\ge\kappa$. The first inequality is DS15 Proposition 4 (exact, every sample, every feasible labeling). Fix $\kappa$; suppose along a subsequence $\mathrm{btw}(\hat s_N;z^{(N)})> v_K-\delta$. Since $\hat v_K\to v_K$, for large $N$ the hypothesis of Lemma DS16.1 holds with $\delta_0=2\delta$. Take $\varepsilon=\varepsilon(\kappa)$ to be fixed below and $\delta=\delta_0(\varepsilon)/2$. The lemma gives, per cell,

$$\bigl|\hat m_{\lambda,b}(z^{(N)})-\hat P_N[S_\lambda1_{\hat s_N\in J^*_b}] \bigr| \le\bigl(\hat P_NS_\lambda^2\bigr)^{1/2}\varepsilon^{1/2}$$

(Cauchy–Schwarz over the symmetric difference), while $\hat P_N[S_\lambda1_{\hat s_N\in J^*_b}]\to E[S_\lambda1_{\hat s\in J^*_b}]=0$ — the weighted GC law over the tilted half-plane class absorbs $\hat B^*_N\to B^*$, dominated convergence plus (S)-atomless boundaries give continuity in the tilt, and (L) makes the limit zero. With $\hat W_b\ge w^*_b-2\varepsilon\ge w^*_{\min}/2$ for $\varepsilon$ small,

$$\hat I_{\lambda\lambda}(z^{(N)}) =\sum_b\frac{\hat m_{\lambda,b}^2}{\hat W_b} \le\frac{2K}{w^*_{\min}}\bigl(C\varepsilon^{1/2}+o(1)\bigr)^2,$$

with $C^2=2ES_\lambda^2$, say. Choose $\varepsilon(\kappa)$ so the right side is eventually $<\kappa$: contradiction with $\hat I_{\lambda\lambda}\ge\kappa$. Hence $\limsup\sup\{\mathrm{btw}\}\le v_K-\delta(\kappa)$ with $\delta(\kappa)=\delta_0(\varepsilon(\kappa))/2$. The (M3) transfer is the diagonal bound $\lambda_{\min}(\hat I_N)\le e_\lambda^\top\hat I_Ne_\lambda =\hat I_{\lambda\lambda}$. ∎

Proof of (2). $\hat\Phi_s(z^{(N)})\to v_K$ forces $\mathrm{btw}(\hat s_N;z^{(N)})\to v_K$ (Proposition 4 sandwich with $\hat v_K\to v_K$). Apply Lemma DS16.1 with $\varepsilon_m\downarrow0$ along $\delta_m\downarrow0$ and diagonalize: symmetric differences to $J^*$-cells vanish, so masses converge to $w^*_b$ (min mass positive, singletons die), and the Step-3 display above with $\varepsilon\to0$ gives $\hat m_{\lambda,b}\to0$, hence $\hat I_{\lambda\lambda}\to0$; the cross block $\hat I_{\psi\lambda}=\sum_b\hat m_{\psi,b}\hat m_{\lambda,b}/\hat W_b\to0$ since $\hat m_{\psi,b}$ is bounded and masses are bounded below; $\lambda_{\min}\le\hat I_{\lambda\lambda}\to0$. ∎

Proof of (3). For a fixed $q$, label by $q(S_i)$, not by applying an arbitrary discontinuous rule to the sample-centered row. Masses and raw moments are sample means of fixed integrable functions. Exact centering only changes the empirical cell moment by

$$\hat m_b=P_N[S1_{\{q(S)=b\}}]-\bar S_NP_N(q(S)=b),$$

which converges to $m_b$ because $\bar S_N\to ES=0$. Thus the LLN gives $\hat W_b\to W_b>0$, $\hat m_b\to m_b$, so $\hat I_{\lambda\lambda}\to I_{q,\lambda\lambda}>\kappa$ (eventually $\ge\kappa$), feasibility eventually, and $\hat\Phi_s\to\Phi(q)$ by continuity of the Schur value at a nonsingular scalar nuisance block. Taking $q$ near the sup gives the floor; the cap of (1) applied to these labelings gives $v^{*+}(\kappa)\le v_K-\delta(\kappa)$. Neither this approximation nor the price proof supplies an optimizer at the strict boundary or continuity in $\kappa$. ∎

Self-adversarial notes (protocol G). Duplicate atoms and split duplicates are inside the grouping formulation by construction; singleton cells are handled by the mass step, not excluded; $\hat I_{\lambda\lambda}\ge\kappa$ excludes singular nuisance blocks but not $\det\hat I_N=0$ (such states have $\hat\Phi_s=0$, inside the cap trivially); the statement is for exact scores and equal weights — estimated scores are “Estimated scores and classifiers”, unequal weights were probed adversarially at finite $N$ (census suite) but are not claimed; the law class needs (S) (atomless, unique scalar optimum) — the census grid laws are atomic emulations, evidence only; nothing here claims margin-carrying exchange-stable sequences exist asymptotically (that inhabitation question is OPEN-DS-STABLE-BASINS). Applying $q$ to centered rows instead of raw $S_i$ is a different, sample-dependent labeling convention and needs boundary/translation continuity not assumed here. The PRICE statement is nevertheless pathwise uniform over all labelings, so it covers either convention whenever the margin is observed.

Measured: which regime the optimizer actually occupies

(N-DS-STABLE-CENSUS, N-DS-STABLE-ASCENT, N-DS-STABLE-LIBRARY; py/ds_stable_margins.py, integer-exact full-lattice census selftested against the audit stack's from-scratch oracle on 782 states.)

• Census (exact, $N=10$–$14$, $K=3$, centered06 + mix3, 2 reps). Exchange-stable states are plentiful (5–944 per instance) and overwhelmingly non-global. On the centered law they span both regimes: $\hat I_{\lambda\lambda}$ from $10^{-5}$ to $0.57$, with gap and nuisance block anti-correlated (per-instance correlation $-0.27$ to $-0.83$) — the finite shadow of conclusion (1). Whenever $\hat I_{\lambda\lambda}>0.2$, the relative gap exceeded $1.2\%$–$8.4\%$; margin-retaining non-global stable states existed in every instance (witness fixture CE-DS-STABLE-MARGIN-RETAINING-001, $N=8$: $\hat I_{\lambda\lambda}\approx0.523$, $\lambda_{\min}\ge0.1397$, min mass $1/4$, separation $0.325$, price $7.7\%$). The DS13 leverage bound held at every spot-checked stable state; the exact sandwich held at all 3,155 stable states censused. (M5) caution: stable states with nearly coincident projected centroids occur (min separation $5.5\times10^{-4}$), so separation must be checked, never assumed (cf. DS12 wasted cells).
• The documented initializer is not a terminal state. The efficient-score DP interval labeling is exchange-unstable in 7/10 centered-law census instances (and already at $N=8$: fixture CE-DS-INTERVAL-SEED-UNSTABLE-001, exact gain $0.447$); the improving move grows the nuisance block 27-fold — near the seed, the profiled objective climbs by buying back nuisance information, the finite face of DS15 Proposition 6's steering. On mix3 the same labeling was stable in 9/10 instances.
• Exact ascent ($N\le14$). Terminal states from the interval seed and 20 random seeds per instance are strongly seed-dependent on the centered law (terminal $\hat I_{\lambda\lambda}$ spanning $0.0002$–$0.57$); margin-retaining terminals are reached with sizable probability at these tiny $N$.
• Library scale ($N=100/300/1000$, gauss06 vs mix3; efficient-score vs k-means++ vs random seeding; optimize_partition + ProfiledDOptimality + DExchangeConfig, exchange-stable terminals verified). On the centered law, every seeding's terminal state collapses the nuisance block at the predicted $K/N$ scale ($N\cdot\hat I_{\lambda\lambda}$ median $0.5$–$3.0$ across all seeds and sizes) with near-optimal values (the research run's reported aggregate log-gap summaries to the certified DP ceiling are $0.004$–$0.046$; this is not a bound on every seed/run), masses approaching the population values and healthy separations — exactly conclusion (2) in action: the terminals are in the funnel because they are near-optimal in value, seed-independently. On mix3 every seeding lands at $\lambda_{\min}\approx1.7=\Theta(1)$ with log-gaps $\le2\times10^{-4}$: the non-centered regime keeps its margins at the optimum, as OPEN-DS-MARGINS-NONCENTERED's first branch conjectures.

Verdict: the compile rule (DS-PROFILED-COMPILE-CERTIFICATE)

On the conditionally centered class, for what the free optimizer returns:

• The projected efficient-score interval rule remains the only currently established unconditional compile path in the registered theory. This is an inventory of proved routes, not a mathematical uniqueness theorem about every possible compiler. Free profiled optimization is value-driven; conclusion (2) makes every value-successful run — global or not, however seeded — nuisance-degenerate in the limit, with the companion rule's slope $\hat B_z$ unstable (0/0). Nothing about the terminal state is stably compilable, and compile_quantizer's refusal for profiled criteria is the correct default, now backed by a theorem about the optimizer's actual output rather than only about global optima.
• DS14 supplies a sequence-conditional companion rule, not a population guarantee from one finite certificate. If an exchange-stable sequence satisfies DS14's full hypotheses — (M1)/(M4) on the law and, eventually, uniform (M2)/(M3)/(M5), or the audited merged-rule variant — then the companion efficient-Voronoi rules $\rho_N$ built from the states' own projected centroids and profiled metrics geometrize and have population-stationary subsequential limits. At one finite $N$, exact exchange stability plus measured mass, conditioning, and separation checks are useful diagnostics and define a candidate companion rule, but do not by themselves certify those asymptotics; DS13 is a consequence of stability, not a replacement for it. DS16 prices any state with measured $\lambda_{\min}(\hat I_N)\ge\kappa$ pathwise: report the observable finite gap $\hat v_K-\hat\Phi_s$ (the efficient_score_bound DP ceiling minus the terminal value). The population $\delta(\kappa)$ is existential and may not be reported numerically without a law-specific lower bound. Measured: on the centered class the free optimizer does not return such states at realistic $N$ (the funnel), so the candidate path operationally appears to require margin-constrained optimization — a solver-design question (an earlier, since-closed question), whose asymptotic non-vacuity (do margin-compatible stable sequences exist? is $v^*(\kappa)$ attained?) is the explicit open remainder OPEN-DS-STABLE-BASINS (OPEN-DS-STABLE-BASINS).
• Information-loss implication. A margin-certified compile retains at most $v_K-\delta(\kappa)$ of profiled information against the $K$-cell ceiling $v_K$ (relative retention $\le1-\delta(\kappa)/v_K$), on top of the binning loss $v_K<\sigma_s^2$ against unbinned inference; the projected rule attains the ceiling only in the distinct projected-score formulation where nuisance information is supplied externally/unbinned (DS9/DS15 deployability). The theorem gives no positive computable lower bound on $\delta(\kappa)$ from a single finite dataset.

Both study stop conditions that could close the deployment question are thus hit: proved (price + funnel on the stated class) and reduced (the inhabitation/selection remainder is OPEN-DS-STABLE-BASINS), with the falsification half serialized (CE-DS-STABLE-MARGIN-RETAINING-001, CE-DS-INTERVAL-SEED-UNSTABLE-001).