Split-class screened supply - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: remaining problems; the reading order is on the full proofs page.

Overview. This dossier proves Proposition 29.3 in total-budget form, on the regular split class (Theorem D25.1). For a product of isotropic one-dimensional log-concave laws and a cut depending on $k$ coordinates, the screened weighted excess over $[0,T\wedge\tau_\eta]$ is at most $(2k+64\eta^2(1+k))/\kappa$ ((D25.4)), uniformly in $T$ , $n$ and the spectators. It is not a $C(k)\,T$ estimate. On the screen the weighted excess is dominated by the Stein source $Q_t$ , and $Q_t$ is then paid from the coordinate source and dissipation budgets. The passage to general split laws is left open (Remark D25.3).

On the screened set, $e_tW_{\rm cut}\le Q_t/\kappa$ pathwise ((D25.5)). This uses only $e_t\ge0$ and gives (D25.6).
In the tight window, Lemma 32.1 converts $Q_t$ into $2S_t+64\eta^2D_t$ ((D25.8)).
Lemma 31.1 and Corollary 24.1 give the source budget $\E\int_0^\infty S_t\,\dd t\le k$ ((D25.9)).
Theorem 24.1 with optional stopping and step 3 gives the dissipation budget $1+k$ ((D25.12)). Chaining steps 1–4 proves the theorem.
Separately, Lemma D25.1 extends Corollary 30.1 to a Carleson input with an additive constant. By Grönwall and Doob, the mass survives in the window up to a time $T_*$ with probability at least $1/2$ , and Lemma 30.1 turns this into a perimeter bound. No screened trace companion and no KLS conclusion is claimed (Remark D25.4).

Scope. This dossier proves the screened weighted-excess supply estimate for coordinate cuts on products of isotropic one-dimensional log-concave measures, in total-budget form: the bound is a constant depending only on the number $k$ of active coordinates, the window half-width $\eta$ , and the screening ratio $\kappa$ ; it is uniform in the horizon $T$ , in the ambient dimension $n$ , and in every spectator coordinate. It is not an estimate of the form $C(k)\,T$ , and no claim of that fixed-time or linear-in- $T$ shape is made here. The dossier asserts nothing about the trace-upgrade cluster (Conjecture 29.1, the high-rank part of Conjecture 29.3, Conjecture 31.1); in particular it neither proves nor compares any occupation estimate of that cluster, in compliance with repository constraint 6. A separate auxiliary lemma (Lemma D25.1 below) records the constant-supply extension of Corollary 30.1; it is logically independent of the main proposition and is clearly separated from it.

Regularity and measurability conventions. We adopt the following conventions:

(M1) We work on the usual $\Prob$ -augmented right-continuous filtration of the driving $n$ -dimensional Brownian motion of Eldan’s stochastic localization.

(M2) Regular split class. The main theorem is stated and proved for initial data in the compact-smooth product-preserving regularity class of the certified Riccati-core and excess-audit dossiers (kls-localization-riccati-core.md and kls-excess-audit.md, both under solutions/): each factor $\mu^{(i)}$ is a compactly supported one-dimensional log-concave law with smooth density, normalized to mean zero and variance one, and the cut $E=E_J\times\R^{J^c}$ is measurable with respect to the coordinates in $J$ with $E_J\subset\R^{J}$ having $C^2$ relative boundary with a tubular neighborhood over the support, so that $P_0(E)<\infty$ and the one-sided tube formula identifies the lower outer Minkowski perimeter with weighted surface area. On every bounded time interval all stochastic integrals below are true martingales after the usual bounded stopping, with regularization-independent constants. Passage to general split laws and cuts is discussed, and its unresolved part explicitly flagged, in Remark D25.3 below.

(M3) $P_t=P_t(E)=\mu_t^+(E)$ is the lower outer Minkowski perimeter, realized pathwise as the increasing limit of infima over the fixed countable rational boundary-layer family of bounded continuous mass martingales (Lemma 32.3, construction of solutions/kls-excess-audit.md); as a pathwise monotone limit of countable infima of continuous adapted processes it is progressively measurable. In the regular class the localized density field is jointly measurable and positive, and the moving profile value $I_{\mu_t}(p_t)$ is jointly measurable in $(t,\omega)$ via a fixed countable regular competitor family; hence the excess

e_t \;=\; e_t(E)\;=\;P_t(E)-I_{\mu_t}(p_t)

(D25.1)

is progressively measurable. This is a measurability convention only: no supermartingale or martingale property of the moving profile infimum is asserted (the caveat of Lemma 32.5 is preserved), and no lower bound on $I_{\mu_t}(p_t)$ is used anywhere in this dossier.

(M4) On $\{0<p_t<1\}$ the processes $A_t$ , $K_t$ , $Q_t$ below are continuous and adapted in the regular class, and the map $(A,K)\mapsto\lambda_{\rm cut}(A,K)$ is Borel on supported pairs $\{(A,K):A\succeq0,\ K=P_{\operatorname{Ran}A}\,K\,P_{\operatorname{Ran}A}\}$ (fixed-rank strata together with the Borel Moore–Penrose formula (D13.3) of solutions/lem-lyapunov-stein-duality.md, with the separate convention $\lambda_{\rm cut}(A,0)=0$ ). Hence the weight $W_{\rm cut}(A_t,K_t)$ and the screened indicator $\one_{\mathcal A_{\kappa,t}}=\one_{\{Q_t-\kappa e_tW_{\rm cut}(A_t,K_t)\ge0\}}$ are progressively measurable. Both appear only inside nonnegative Lebesgue-time integrals, handled by Tonelli; no Itô differential is ever taken of the weight, of the indicator, or of any function of $\lambda_{\rm cut}$ .

Notation. Fix the manuscript localization notation: $p_t=\mu_t(E)$ , $q_t=1-p_t$ , $s_t=p_tq_t$ , $\delta_t=m_t^E-m_t^F$ , $G_t=\Sigma_t^E-\Sigma_t^F$ , $K_t=G_t+(q_t-p_t)\delta_t\delta_t^T$ , $B_t=s_t\delta_t\delta_t^T$ , $R_t=A_t-B_t$ , $r_t=\Tr B_t=s_t\abs{\delta_t}^2$ , $S_t=s_t\norm{G_t}_\HS^2$ , and $D_t=2s_t\delta_t^TA_t\delta_t-r_t^2\ge r_t^2\ge0$ (Theorem 24.1). Set

Q_t\;=\;s_t\norm{K_t}_\HS^2\;=\;\frac{\calS_{\mu_t}(E)}{s_t},

(D25.2)

the second equality being the certified Stein-norm identity $\calS_\nu(E)=s^2\norm K_\HS^2$ of Proposition 32.1. With the convention $\lambda_{\rm cut}(A,0)=0$ of Lemma 30.3, put

W_{\rm cut}(A,K)\;=\;\bigl(1+\lambda_{\rm cut}(A,K)\bigr)^{5/2}, \qquad \mathcal A_{\kappa,t}\;=\;\bigl\{Q_t\ \ge\ \kappa\,e_t\,W_{\rm cut}(A_t,K_t)\bigr\}.

(D25.3)

Finally, for $\eta\in(0,1/4]$ let $\tau_\eta=\inf\{t\ge0:\abs{p_t-1/2}>\eta\}$ be the continuous-exit time of solutions/kls-localization-riccati-core.md; it is a stopping time for the augmented right-continuous filtration because $p$ has continuous paths.

Refined statement. The theorem below is the verbatim statement of the candidate ledger node prop:split-screened-supply, restricted per convention (M2) to the regular split class; Remark D25.3 records the approximant convention for general split laws together with the explicit unresolved limit-interchange gap.

Theorem D25.1 (split-class screened weighted supply, total-budget form)

Let $\mu=\bigotimes_{i=1}^n\mu^{(i)}$ be a product of isotropic one-dimensional log-concave probability measures in the regular split class (M2), let $J\subset\{1,\dots,n\}$ with $\abs J=k$ , and let $E$ be measurable with respect to the coordinates in $J$ with $0<p_0=\mu(E)<1$ . Then, for every $\eta\in(0,1/4]$ , every $\kappa>0$ , and every $T>0$ ,

\E\int_0^{T\wedge\tau_\eta} e_t\,\bigl(1+\lambda_{\rm cut}(A_t,K_t)\bigr)^{5/2}\, \one_{\{Q_t\ \ge\ \kappa\,e_t\,W_{\rm cut}(A_t,K_t)\}}\,\dd t \;\le\;\frac{2k+64\,\eta^2(1+k)}{\kappa}.

(D25.4)

The right-hand side is independent of $T$ , of the ambient dimension $n$ , and of every spectator factor $\mu^{(i)}$ , $i\notin J$ .

Proof D25.1

Throughout, product structure is preserved pathwise by stochastic localization and the posterior covariance $A_t$ is diagonal (Proposition 27.2); the fixed cut $E$ remains $J$ -measurable; and, since $0<p_0<1$ and the finite-time posterior has a strictly positive likelihood relative to $\mu$ , we have $0<p_t<1$ at every finite time, so all the displayed two-color quantities are defined along the whole path. If $\abs{p_0-1/2}>\eta$ then $\tau_\eta=0$ and (D25.4) is the trivial statement $0\le(2k+64\eta^2(1+k))/\kappa$ ; assume therefore $\abs{p_0-1/2}\le\eta$ from now on. All integrands below are nonnegative and progressively measurable by conventions (M3)–(M4), so every interchange of $\E$ and $\dd t$ is Tonelli for nonnegative integrands and needs no a priori integrability.

Step 1: pathwise domination on the aligned set. We claim that, pathwise, for every $t$ with $0<p_t<1$ ,

e_t\,W_{\rm cut}(A_t,K_t)\,\one_{\mathcal A_{\kappa,t}} \;\le\;\frac{Q_t}{\kappa}.

(D25.5)

First, $e_t\ge0$ : the set $E$ has $\mu_t$ -mass $p_t$ and perimeter $P_t(E)$ , hence is a competitor in the infimum defining the profile value $I_{\mu_t}(p_t)$ , so $I_{\mu_t}(p_t)\le P_t(E)$ . This uses only the definition of the profile as an infimum (the upper-bound direction); no profile lower bound enters. Now, on $\mathcal A_{\kappa,t}$ the defining inequality reads $\kappa\,e_tW_{\rm cut}\le Q_t$ , and dividing by $\kappa>0$ gives $e_tW_{\rm cut}\le Q_t/\kappa$ there; off $\mathcal A_{\kappa,t}$ the left side of (D25.5) vanishes while $Q_t=s_t\norm{K_t}_\HS^2\ge0$ . This proves (D25.5) with no case excluded.

For the record, the degenerate state $K_t=0$ is covered by the conventions of Lemma 30.3 and charges nothing: there $Q_t=0$ and $\lambda_{\rm cut}(A_t,0)=0$ , so $W_{\rm cut}=1$ and membership in $\mathcal A_{\kappa,t}$ reads $0\ge\kappa e_t$ , which (as $e_t\ge0$ ) holds exactly when $e_t=0$ ; in either case both sides of (D25.5) vanish.

Integrating (D25.5) in time and taking expectations,

\E\int_0^{T\wedge\tau_\eta} e_t\,W_{\rm cut}(A_t,K_t)\,\one_{\mathcal A_{\kappa,t}}\,\dd t \;\le\;\frac1\kappa\,\E\int_0^{T\wedge\tau_\eta}Q_t\,\dd t .

(D25.6)

Step 2: tight-window conversion to Riccati currencies. On $\{t<\tau_\eta\}$ we have $\abs{p_t-1/2}\le\eta\le1/4$ , hence $s_t\ge1/4-\eta^2\ge3/16>1/8$ , and the second inequality of the certified conversion Lemma 32.1 applies:

Q_t=\frac{\calS_{\mu_t}(E)}{s_t}\;\le\;2S_t+64\,\eta^2D_t \qquad\text{on }\{t<\tau_\eta\}.

(D25.7)

The single time $t=\tau_\eta$ has Lebesgue measure zero in the time integral, so

\E\int_0^{T\wedge\tau_\eta}Q_t\,\dd t \;\le\;2\,\E\int_0^{T\wedge\tau_\eta}S_t\,\dd t +64\,\eta^2\,\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t .

(D25.8)

Step 3: the coordinate source budget under $0<p_0<1$ . We prove

\E\int_0^\infty S_t\,\dd t\;\le\;\sum_{i\in J}(R_0)_{ii}\;\le\;k .

(D25.9)

This is part (i) of Theorem 31.1, whose certified dossier (solutions/kls-product-covariance.md) records in its dependency audit that this part uses only $0<p_0<1$ and not the packaged coarse balance window $p_0\in[2/5,3/5]$ ; for the reviewer’s convenience we reproduce the three-line certified argument, which cites only unconditional inputs. At every finite time, $(\mu_t,E)$ is a finite-second-moment product pair with a $J$ -measurable cut and $0<p_t<1$ , so Lemma 31.1 gives that $G_t$ is supported on $J\times J$ ; hence its columns $G_te_i$ vanish for $i\notin J$ and

S_t=s_t\norm{G_t}_\HS^2=\sum_{i\in J}s_t\abs{G_te_i}^2 .

(D25.10)

For each fixed $i$ , the unconditional per-direction estimate Corollary 24.1 gives $\E\int_0^Ts_t\abs{G_te_i}^2\,\dd t\le e_i^TR_0e_i=(R_0)_{ii}$ for every $T>0$ . Since $R_0=A_0-B_0\preceq A_0=I_n$ by isotropy of the product, $(R_0)_{ii}\le1$ . Summing over $i\in J$ and letting $T\to\infty$ by monotone convergence proves (D25.9). In particular, since $S\ge0$ ,

\E\int_0^{T\wedge\tau_\eta}S_t\,\dd t\;\le\;k .

(D25.11)

Step 4: the dissipation budget. We prove

\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t\;\le\;1+k .

(D25.12)

By Theorem 24.1 there is a scalar local martingale $M$ with $\dd r_t=\dd M_t+(S_t-D_t)\,\dd t$ . In the regular class (M2), let $(\sigma_m)_{m\ge1}$ be an increasing sequence of bounded stopping times tending to infinity such that each stopped $M^{\sigma_m}$ is a true martingale and all stopped coefficients are integrable — exactly the localizing times used in the certified dossiers solutions/kls-localization-riccati-core.md and solutions/kls-product-covariance.md. Optional stopping at the bounded stopping time $T\wedge\tau_\eta\wedge\sigma_m$ gives

\E\, r_{T\wedge\tau_\eta\wedge\sigma_m} \;=\;r_0+\E\int_0^{T\wedge\tau_\eta\wedge\sigma_m}(S_t-D_t)\,\dd t ,

(D25.13)

whence, discarding $\E\,r_{T\wedge\tau_\eta\wedge\sigma_m}\ge0$ and enlarging the $S$ -integral to $[0,\infty)$ (both integrands are nonnegative),

\E\int_0^{T\wedge\tau_\eta\wedge\sigma_m}D_t\,\dd t \;\le\;r_0+\E\int_0^\infty S_t\,\dd t \;\le\;r_0+k

(D25.14)

by (D25.9). Since $D\ge0$ and $T\wedge\tau_\eta\wedge\sigma_m\uparrow T\wedge\tau_\eta$ , monotone convergence in $m$ gives $\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t\le r_0+k$ . Finally $r_0\le1$ : the covariance decomposition gives $B_0\preceq A_0=I_n$ , and $B_0=s_0\delta_0\delta_0^T$ has rank at most one, so its trace $r_0$ is its only nonzero eigenvalue and is at most 1. This proves (D25.12).

Step 5: combination. Chaining (D25.6), (D25.8), (D25.11), and (D25.12),

\E\int_0^{T\wedge\tau_\eta} e_t\,W_{\rm cut}(A_t,K_t)\,\one_{\mathcal A_{\kappa,t}}\,\dd t \;\le\;\frac1\kappa\Bigl(2k+64\,\eta^2(1+k)\Bigr),

(D25.15)

which is (D25.4). Every bound used — (D25.5), the conversion constants 2 and $64\eta^2$ , the source budget $k$ , and the dissipation budget $1+k$ — is independent of $T$ , of $n$ , and of the spectator factors, which never entered; the constant is therefore total-budget as claimed.

Remark D25.1 (Shape of the bound; what is not claimed)

The bound (D25.4) is a total budget: a single constant $c_E(k,\eta,\kappa)=(2k+64\eta^2(1+k))/\kappa$ controlling the screened weighted excess over the entire window $[0,T\wedge\tau_\eta]$ for every $T>0$ simultaneously. It is not of the form $C(k)\,T$ , and this dossier does not assert any fixed-time expected-source estimate $\sup_{t\le T_0}\E[Q_t\one_{\mathcal A_{\kappa,t}}\one_{\{t<\tau_\eta\}}]\le C(k)$ that would yield such a form; that upgrade is recorded as open in the probe record. Likewise nothing here bears on the trace-upgrade cluster (Conjecture 29.1, high-rank Conjecture 29.3, Conjecture 31.1) or on the general (non-product) screened supply: no occupation estimate, equivalence, or comparison across that cluster is proved or implied (constraint 6).

Remark D25.2 (Interpretation: the weight is cut-local on the split class)

This remark is interpretive and is used nowhere in the proof of Theorem D25.1. Pathwise, $A_t$ is diagonal (Proposition 27.2) with positive diagonal entries $A_t^{(i)}$ in the regular class, and $K_t=P_JK_tP_J$ by Lemma 31.1. By the certified eigenbasis identity (D13.17) of solutions/lem-lyapunov-stein-duality.md (Remark on the auxiliary harmonic-mean formula), $\lambda_{\rm cut}(A_t,K_t)$ is the $\abs{(K_t)_{ij}}^2$ -weighted harmonic mean of $(A_t^{(i)}+A_t^{(j)})/2$ over $(i,j)\in J\times J$ , hence

\lambda_{\rm cut}(A_t,K_t)\;\le\;\max_{i\in J}A_t^{(i)} :

(D25.16)

on the split class the weight sees only the $J$ -block covariance scale; spectator covariance spikes are invisible to both $Q_t$ and $W_{\rm cut}$ , consistently with the uniformity in $n$ and in the spectators of the theorem.

Remark D25.3 (General split laws: approximant convention and explicit gap)

For a general product $\mu=\bigotimes_i\mu^{(i)}$ of isotropic one-dimensional log-concave laws and a general $J$ -measurable cut with $0<p_0<1$ and $P_0(E)<\infty$ , statement (D25.4) is defined on approximants, per convention (M2) and Section 5(5) of the probe record: apply Theorem D25.1 to the coordinatewise product-preserving truncated and smoothed approximants used in the certified scheme of solutions/kls-product-covariance.md (each factor truncated, smoothed, and renormalized to isotropic position — which preserves the class — and the cut regularized within the $J$ -coordinates, preserving $J$ -measurability); the constant $(2k+64\eta^2(1+k))/\kappa$ is uniform over the approximation. What is not proved here, and is flagged as an explicit open technical step, is the limit interchange for the screened integrand itself: the indicator $\one_{\mathcal A_{\kappa,t}}$ does not converge monotonically along the approximation, so the left side of (D25.4) evaluated directly on the limiting rough data is not obtained from the approximant bound by Fatou. Any consumer of the general-class statement must either work with the approximants or close this interchange separately.

Auxiliary lemma: constant-supply extension of tight-window consumption¶

The following lemma is not part of Theorem D25.1 and is not needed by its proof. It extends the certified Corollary 30.1 by allowing a constant additive term in the absorptive Carleson input — exactly the shape a total-budget supply such as (D25.4) would feed, as recorded in Section 1 of the probe. It is stated in the general (not split-specific) localization setting of the certified Riccati dossier, in the regular class of (M2) with its regularization-independent constants.

Lemma D25.1 (tight-window consumption with a constant supply term)

Let $\mu$ be isotropic and log-concave, $E$ a cut, and fix $0<\eta\le1/6$ with $\abs{p_0-1/2}\le\eta/2$ . Suppose that for constants $c,C_0,C_1\ge0$ , a number $\alpha<1$ , a horizon $T_0>0$ , and every $T\le T_0$ ,

\E\int_0^{T\wedge\tau_\eta}S_t\,\dd t \;\le\;c+C_0T +C_1\,\E\int_0^{T\wedge\tau_\eta}r_t\,\dd t +\alpha\,\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t ,

(D25.17)

and that

\E\int_0^{T_0\wedge\tau_\eta}D_t\,\dd t<\infty

(D25.18)

(automatic in the regular class (M2)). Set $C_*=(1+c+C_0T_0)\,e^{C_1T_0}$ . Then for every positive $T_*\le\min\bigl(T_0,\ 4\eta^2/(9C_*)\bigr)$ ,

\Prob\bigl\{\min(p_{T_*},q_{T_*})\ge1/3\bigr\}\;\ge\;\frac12 ,

(D25.19)

and hence Lemma 30.1 gives $\mu^+(E)\ge c'\sqrt{T_*}$ for a numerical $c'>0$ , with $T_*$ depending only on $(T_0,c,C_0,C_1,\eta)$ .

Proof D25.2

This is the proof of the certified Corollary 30.1 (Section 4 of solutions/kls-localization-riccati-core.md) with one line changed in the Gronwall input; we run it in full. Let $u(T)=\E\,r_{T\wedge\tau_\eta}$ for $0\le T\le T_0$ . In the regular class, $r_t=\Tr B_t\le\Tr A_t$ is bounded on bounded intervals by the squared support diameter, so $u$ is finite and bounded on $[0,T_0]$ , and it is measurable ( $r$ has continuous paths, so $u$ is a pointwise limit of continuous functions by dominated convergence); the constants produced below do not depend on the regularization.

Integrate Theorem 24.1 at the bounded stopping times $T\wedge\tau_\eta\wedge\sigma_m$ , with $(\sigma_m)$ the localizing sequence of (M2), and let $m\uparrow\infty$ : Fatou on the left (as $r\ge0$ has continuous paths) and monotone convergence for the two nonnegative occupation terms give

u(T)\;\le\;r_0+\E\int_0^{T\wedge\tau_\eta}S_t\,\dd t -\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t .

(D25.20)

Insert (D25.17) and use $r_0\le1$ (rank-one $B_0\preceq A_0=I$ ). The term $(\alpha-1)\E\int_0^{T\wedge\tau_\eta}D_t\,\dd t$ is nonpositive and finite by (D25.18) with $\alpha<1$ , so it may be discarded — this is the one place the absorption margin and the a priori finiteness are used — leaving

u(T)\;\le\;1+c+C_0T +C_1\,\E\int_0^{T\wedge\tau_\eta}r_t\,\dd t \;\le\;1+c+C_0T+C_1\int_0^Tu(t)\,\dd t ,

(D25.21)

the last inequality by the pathwise bound $\one_{\{t<\tau_\eta\}}r_t\le r_{t\wedge\tau_\eta}$ and Tonelli. Gronwall’s inequality for the bounded measurable $u$ yields

u(T)\;\le\;(1+c+C_0T)\,e^{C_1T}\;\le\;C_* \qquad(0\le T\le T_0).

(D25.22)

Before exit, $\abs{p_t-1/2}\le\eta\le1/6$ gives $s_t\ge1/4-\eta^2\ge2/9$ , hence $\abs{\delta_t}^2=r_t/s_t\le\tfrac92 r_t$ and

\E\int_0^{T\wedge\tau_\eta}\abs{\delta_t}^2\,\dd t \;\le\;\frac92\int_0^T\E\bigl[\one_{\{t<\tau_\eta\}}r_t\bigr]\,\dd t \;\le\;\frac92\,C_*T .

(D25.23)

Since $\abs{p_0-1/2}\le\eta/2$ , continuity of $p$ forces $\sup_{t\le T\wedge\tau_\eta}\abs{p_t-p_0}\ge\eta/2$ on $\{\tau_\eta\le T\}$ . Doob’s $L^2$ inequality applied to the stopped mass martingale, whose quadratic variation is $\dd[p]_t=s_t^2\abs{\delta_t}^2\,\dd t$ with $s_t\le1/4$ , gives

\Prob(\tau_\eta\le T) \;\le\;\frac4{\eta^2}\,\E[p]_{T\wedge\tau_\eta} \;\le\;\frac{4}{16\,\eta^2}\cdot\frac92\,C_*T \;=\;\frac{9C_*T}{8\eta^2}.

(D25.24)

For $T_*\le\min(T_0,4\eta^2/(9C_*))$ this is at most $1/2$ , and on $\{\tau_\eta>T_*\}$ we have $\min(p_{T_*},q_{T_*})\ge1/2-\eta\ge1/3$ . The boundary consequence is Lemma 30.1 with $(T_0,c_0,b_0)=(T_*,1/2,1/3)$ .

Obstructions respected.

rem:two-tail-slice-bounds (Proposition 25.2): that fence kills slice-wise absolute-scale Stein estimates and calibrates the necessary covariance-weight power $5/2$ . Theorem D25.1 is a time-integrated expectation bound, not a slice-wise estimate; it retains exactly the calibrated weight $(1+\lambda_{\rm cut})^{5/2}$ , whose two-tail value is $\lambda_{\rm cut}(A_\Lambda,K_\Lambda)=\Lambda$ by the certified calibration (D13.20), so no weaker power is smuggled in; and the two-tail initial laws $N(0,\diag(\Lambda,1,\dots,1))$ , $\Lambda>1$ , lie outside the hypothesis class (their first factor is not isotropic). States of two-tail type reached dynamically remain chargeable by the screen and are paid for through the budget $Q_t/\kappa$ ; the theorem never discards them.
rem:profile-circularity: no lower bound on the moving profile $I_{\mu_t}(p_t)$ — and no profile information beyond its definition as an infimum, used once in Step 1 in the upper-bound direction $I_{\mu_t}(p_t)\le P_t(E)$ to record $e_t\ge0$ — enters any step. The excess-identity refinement fenced by this obstruction is not used; indeed $e_t$ is touched only through the pathwise screen inequality (D25.5).