The occupation implication

Part of the fixed-eigenfunction mechanism, Chapter The fixed eigenfunction: following one eigenfunction through localization; the reading order is on the full proofs page.

Overview. This dossier gives the occupation implication Theorem D22.1 with constants $C_0=34$ , $C_1=0$ on $[0,T_0(n)]$ , under the small-gap hypothesis $\lambda\le3/(8K_n)$ . With the stated input Theorem 26.2, this hypothesis has no admissible measure: $\lambda=1/\CP(\mu)\ge1/K_n>3/(8K_n)$ . The implication therefore supplies no nonempty case of Conjecture 21.1. The bound $\CP\le C\log^2n$ in Theorem D22.2 already follows from the stronger published input. Separately, Lemma 21.3 bounds the source before covariance exit for fixed unit-variance tests without the small-gap restriction. The calculations below record the stopping, restart, optional-projection and fixed-dimensional bridge estimates with their constants.

The exit time $\tau=\tau_2$ of $\norm{A_t}_\op$ from level 2 is an a.s. positive stopping time (Lemma D22.1), and the imported window bound Theorem 26.1 transfers to the planted realization (Lemma D22.2).
Optional projection bounds $\E[v_\sigma^2\one_E]$ by $\E_\mu f^4$ (Lemma D22.3); a tail integral bounds $\E[\tau^{-2}\one_{\{\tau\le T\}}]$ using step 1 (Lemma D22.4), which fixes the universal cap $t_c$ (Definition D22.1).
The source is split at $\tau$ : the pre-exit part costs $32T$ by the stopped-window input, and the post-exit part costs $\sqrt2\,T$ by the restart input, Cauchy–Schwarz, step 2 and the small-gap bound $\E_\mu f^4\le2$ . Together these give (D22.13).
On regular approximants, the fixed- $n$ bridge ( $q$ -identity, terminal variance, posterior Brascamp–Lieb) with step 3 gives $\lambda\ge T_*(n)/2$ on the small-gap branch; the large-gap branch is immediate. This yields (D22.31).
The certified approximation passage at fixed dimension carries the bound to every isotropic log-concave law.

Refined statement. Using Theorem 25.1 as a proved dependency, this dossier proves that the occupation hypothesis of Conjecture 21.1 holds on the covariance window $[0,T_0(n)]$ , $T_0(n)=\min\{t_c,1/(\bar C\log^2n)\}$ , with constants $C_0=34$ and $C_1=0$ and no damping consumed, for every first eigenfunction on the small-gap branch $\lambda\le3/(8K_n)$ ; and that, combined with the certified bridge argument of solutions/prop-spectral-sufficiency.md rerun at fixed $n$ and the trivial large-gap branch, every isotropic log-concave probability on $\R^n$ ( $n\ge2$ ) satisfies

\CP\ \le\ C\log^2n,

(D22.1)

with $C$ universal.

Scope fences (stated up front). This dossier does not claim, and must not be read as claiming:

Not the gate. Conjecture 21.1 demands universal constants $T_0,C_0,C_1$ ; here $T_0(n)\to0$ as $n\to\infty$ . The question remains open and its status is unchanged by this dossier.
No universal-time statement and no beyond-window bound. Nothing is asserted for $t>T_0(n)$ . By the imported Proposition 0.1, the exit event $\{\tau_2\le t\}$ ceases to be rare past times of order $1/\log n$ , so the rarity mechanism used here provably cannot be extended; this dossier does not attempt it.
Empty small-gap branch and redundant frontier consequence. The published input Theorem 26.2 gives $\CP(\mu)\le K_n$ , hence $\lambda\ge1/K_n>3/(8K_n)$ for every regular isotropic approximant. Thus the small-gap occupation implication has an empty admissible class. Every actual law falls in the large-gap branch, and the conclusion $\CP\le C\log^2n$ is weaker than the input $\CP\le C_K\log n$ . These calculations provide no evidence that the occupation estimate holds for any admissible eigenfunction. The separate stopped-source estimate Lemma 21.3 does not impose the small-gap restriction.
Nothing about the trace-upgrade cluster. No statement is made or implied about Conjecture 29.1, Conjecture 29.3, or conj:product-alignment (the brief’s trace-upgrade comparison constraint).

Setting. Throughout, $\mu$ is a regular isotropic approximant on $\R^n$ , $n\ge2$ , in the class of Section Setup and the exact fixed-function SDE: $\dd\mu=Z^{-1}e^{-V}\dd x$ with $V\in C^\infty$ , $\nabla^2V\succeq\varepsilon I_n$ for some $\varepsilon>0$ , centered and isotropic, with Friedrichs generator $L=\Delta-\nabla V\cdot\nabla$ of compact resolvent, and $f$ a normalized first nonconstant eigenfunction, $-Lf=\lambda f$ , $\E_\mu f=0$ , $\E_\mu f^2=1$ . The localization is realized by the planted channel $c_t=tX+B_t^{\mathrm{obs}}$ with $X\sim\mu$ and $B^{\mathrm{obs}}$ an independent Brownian motion; $(\mathcal F_t)$ is the usual augmentation of the observation filtration, and the pathwise posterior kernel $\mu_t$ , the quantities $m_t,a_t,v_t,A_t,g_t,H_t$ , and the zero-convention for $H_t$ are exactly as in the companion restart dossier (Theorem D15.1). Write $S_t=\norm{H_t}_{\HS}^2$ , $q(t)=\E\abs{g_t}^2$ , $D_t=g_t^TA_tg_t\ge0$ , and

\tau_L=\inf\{t\ge0:\norm{A_t}_\op\ge L\},\qquad \tau:=\tau_2 .

(D22.2)

Inputs and their exact standing.

(I1) Theorem 25.1 Letwin, 2026, Thm. 1.2: $\Var(Y^TMY)\le8\norm M_{\HS}^2$ for isotropic log-concave $Y$ and symmetric $M$ . This proved dependency applies to all isotropic log-concave laws, including every whitened posterior used in (I4), without an extra bounded-support restriction.

(I2) Theorem 26.1 Klartag & Lehec, 2025, Thm. 61: there is a universal $\bar C$ such that for every isotropic log-concave $\mu$ on $\R^n$ and every $t\le t_1(n):=1/(\bar C\log^2n)$ , $\Prob(\exists s\le t:\norm{A_s}_\op\ge2)\le e^{-1/(\bar Ct)}$ . Imported, published. Without loss of generality $\bar C\ge1$ : enlarging $\bar C$ shrinks the window and weakens the bound, so the statement with any $\bar C$ implies the statement with $\max\{\bar C,1\}$ .

(I3) Theorem 26.2 Klartag, 2023 (published import): every isotropic log-concave probability on $\R^n$ , $n\ge2$ , satisfies $\CP\le K_n:=C_K\log n$ with $C_K$ universal, via the Cheeger bound $\psi_n\le C\sqrt{\log n}$ and Cheeger’s inequality. Cited as an import; not proved here.

(I4) Companion dossier lem-mm-stopped-window-source.md in solutions/ (Theorem D17.1; uses the certified (I1)): for every regular approximant, fixed unit-variance $f\in L^2(\mu)$ , $L\ge1$ , $T>0$ : $\E\int_0^{T\wedge\tau_L}S_t\dd t\le8L^2T$ .

(I5) Companion dossier lem-mm-restart-deweighting.md in solutions/ (Theorem D15.1; unconditional): for every a.s.\ positive stopping time $\sigma$ , a.s. on $\{\sigma<\infty\}$ , $\E[\int_\sigma^\infty S_t\dd t\mid\mathcal F_\sigma] \le\Var_{\mu_\sigma}(f)/(\varepsilon+\sigma)\le v_\sigma/\sigma$ , together with the optional-time identification $\int\phi\dd\mu_\sigma=\E[\phi(X)\mid\mathcal F_\sigma]$ a.s. on $\{\sigma<\infty\}$ for every measurable $\phi\ge0$ , and the deterministic solution map $\mathsf S_\mu$ of the observation equation with its pathwise uniqueness.

(I6) Companion dossier lem-mm-smallgap-fourth-moment.md in solutions/ (Theorem D16.1; uses (I3)): if $\lambda\le3/(8K_n)$ then $\E_\mu f^4\le2$ .

(I7) Certified nodes: Lemma 21.2 (consumed inside (I5)) and the certified bridge dossier for Proposition 21.1, namely the file prop-spectral-sufficiency.md in solutions/, whose internal steps ( $q$ -identity, Bessel bound, terminal-variance identity, posterior Brascamp–Lieb, and the fixed-test approximation passage) are rerun here at fixed $n$ ; the certified theorem itself is not invoked as a black box, because its hypothesis requires dimension-free constants.

Inputs (I4)–(I6) are the companion results Lemma 21.3, Lemma 21.4, and Lemma 21.5. Their detailed dossiers provide the stated forms; the stopped-source result now uses the certified quadratic theorem as a proof dependency. This dossier’s own contribution is the assembly: the exit-time bookkeeping, the optional projection step, the tail integral, the constant audit, and the fixed- $n$ rerun of the bridge.

Preliminary lemmas¶

Proof D22.1

On any compact time interval $\sup_{s\le U}\abs{c_s}<\infty$ pathwise, and the entries of $A_t$ are ratios of integrals of $(1+\abs x^2)e^{c_t\cdot x-t\abs x^2/2}$ -type integrands against $\mu$ ; dominated convergence (domination by $(1+\abs x^2)e^{R\abs x}\in L^1(\mu)$ , by strong log-concavity) gives continuity of $t\mapsto A_t$ , hence of $t\mapsto\norm{A_t}_\op$ . (i): by continuity, $\{\tau_L\le t\}=\{\sup_{s\in([0,t]\cap\mathbb Q)\cup\{t\}}\norm{A_s}_\op\ge L\} \in\mathcal F_t$ . (ii): $A_0=\Cov(\mu)=I_n$ by isotropy, so $\norm{A_0}_\op=1<2$ and continuity forces $\tau>0$ . (iii): if $\tau\le t$ then continuity gives $\norm{A_\tau}_\op\ge2$ at a time $\le t$ ; conversely if $\norm{A_s}_\op\ge2$ for some $s\le t$ then $\tau\le s\le t$ . (iv): $\{\tau\le T\}\cap\{\tau\le t\} =\{\tau\le T\wedge t\}\in\mathcal F_{t}$ for every $t$ , which is the definition of membership in $\mathcal F_\tau$ .

Proof D22.2

Theorem 26.1 is a statement about the law of $\sup_{s\le t}\norm{A_s}_\op$ for the stochastic localization of $\mu$ as defined in Section The process: the tilt density (2.1) driven by the tilt process solving $\dd c_t=a_t\dd t+\dd W_t$ for a standard Brownian motion $W$ , $c_0=0$ . For the $\varepsilon$ -strongly log-concave $\mu$ , the drift $c\mapsto\mathsf a(\mu;t,c)$ of this equation is globally Lipschitz uniformly in $t$ (Brascamp–Lieb, constant $\varepsilon^{-1}$ ), so by the deterministic solution map of the companion restart dossier (Theorem D15.1, Step 6), every solution driven by any standard Brownian motion is the pathwise image $\mathsf S_\mu(W)$ of its driver; its law is therefore the fixed push-forward $\mathsf S_\mu\#\mathbb W$ of Wiener measure, the same for every realization. The planted channel realizes this equation with the innovation Brownian motion (Theorem D15.1, Steps 4 and 7(a), at $\sigma=0$ ), and the posterior kernel equals the tilt density (2.1) evaluated on the tilt path. Hence the law of $(A_s)_{s\ge0}$ — a fixed measurable functional of the tilt path — is identical in the two realizations, and the imported bound transfers verbatim. Combining with Lemma D22.1(iii) gives the display.

Proof D22.3

By the optional-time identification of the companion restart dossier ((I5), applied with the nonnegative integrable test functions $f^2$ and $f^4$ ), almost surely on $\{\sigma<\infty\}$ ,

\int f^2\dd\mu_\sigma=\E\bigl[f^2(X)\mid\mathcal F_\sigma\bigr], \qquad \int f^4\dd\mu_\sigma=\E\bigl[f^4(X)\mid\mathcal F_\sigma\bigr].

(D22.5)

Pathwise, $0\le v_\sigma\le\int f^2\dd\mu_\sigma$ , and by conditional Jensen ( $Z=f^2(X)$ ),

v_\sigma^2\le\Bigl(\E\bigl[f^2(X)\mid\mathcal F_\sigma\bigr]\Bigr)^2 \le\E\bigl[f^4(X)\mid\mathcal F_\sigma\bigr] \quad\text{a.s.\ on }\{\sigma<\infty\}.

(D22.6)

Multiplying by $\one_E$ (which is $\mathcal F_\sigma$ -measurable) and taking expectations, the tower property gives the first inequality; the second is trivial.

Proof D22.4

Write $F(s)=\Prob(\tau\le s)$ ; by Lemma D22.2, $F(s)\le e^{-1/(\bar Cs)}$ for $0<s\le T\le t_1(n)$ , and $\tau>0$ a.s. by Lemma D22.1(ii). For $0<\delta<\tau\le T$ one has the calculus identity $\tau^{-2}=T^{-2}+2\int_\tau^Ts^{-3}\dd s$ , so by Tonelli’s theorem

\E\bigl[\tau^{-2}\one_{\{\delta<\tau\le T\}}\bigr] =T^{-2}\,\Prob(\delta<\tau\le T) +2\int_\delta^Ts^{-3}\,\Prob(\delta<\tau\le s)\dd s \le T^{-2}F(T)+2\int_0^Ts^{-3}F(s)\dd s .

(D22.8)

Letting $\delta\downarrow0$ by monotone convergence bounds the left-hand side of the lemma by the right-hand side above. Substituting $u=1/s$ ,

\int_0^Ts^{-3}e^{-1/(\bar Cs)}\dd s =\int_{1/T}^\infty u\,e^{-u/\bar C}\dd u =\Bigl(\frac{\bar C}{T}+\bar C^2\Bigr)e^{-1/(\bar CT)},

(D22.9)

using $\int_a^\infty ue^{-u/\bar C}\dd u=(a\bar C+\bar C^2)e^{-a/\bar C}$ . Hence

\E\bigl[\tau^{-2}\one_{\{\tau\le T\}}\bigr] \le\Bigl(T^{-2}+\frac{2\bar C}T+2\bar C^2\Bigr)e^{-1/(\bar CT)} .

(D22.10)

Finally, for $T\le1$ and $\bar C\ge1$ : $1+2\bar C+2\bar C^2\le5\bar C^2$ , because $3\bar C^2-2\bar C-1=(3\bar C+1)(\bar C-1)\ge0$ ; so the bracket is at most $5\bar C^2T^{-2}$ .

Definition D22.1 (The universal window cap

t_c

)

Let $h(t)=\sqrt5\,\bar C\,t^{-2}e^{-1/(2\bar Ct)}$ for $t>0$ . Then $h(t)\to0$ as $t\downarrow0$ , and $\frac{\dd}{\dd t}\log h(t) =t^{-1}\bigl(\tfrac1{2\bar Ct}-2\bigr)>0$ for $t<1/(4\bar C)$ , so $h$ is continuous and increasing on $(0,1/(4\bar C)]$ . Define

t_c\ :=\ \sup\bigl\{t\in(0,\tfrac1{4\bar C}]:\ h(t)\le1\bigr\}\ \in\ (0,1).

(D22.11)

By monotonicity and continuity, $h(t)\le1$ for every $t\in(0,t_c]$ , and $t_c\le1/(4\bar C)<1$ . The number $t_c$ depends only on the universal constant $\bar C$ , hence is universal.

The window occupation estimate¶

Theorem D22.1 (Window occupation with

C_0=34

C_1=0

)

Set

T_0(n)=\min\Bigl\{t_c,\ \frac1{\bar C\log^2n}\Bigr\} .

(D22.12)

Then for every regular isotropic approximant $\mu$ on $\R^n$ ( $n\ge2$ ), every normalized first eigenfunction $f$ with $\lambda\le\dfrac3{8K_n}$ , and every $T\le T_0(n)$ ,

\E\int_0^TS_t\dd t\ \le\ 34\,T .

(D22.13)

In particular the occupation hypothesis (21.11) of Conjecture 21.1 holds on $[0,T_0(n)]$ with

C_0=34,\qquad C_1=0,

(D22.14)

and with no damping consumed: the nonnegative terms $C_1\int_0^Tq+2\E\int_0^TD_t\dd t$ are not needed on the right-hand side.

Proof D22.5

Fix $T\le T_0(n)$ . Since $f\in L^2(\mu)$ with $\Var_\mu(f)=\E_\mu f^2=1$ ( $\E_\mu f=0$ ), input (I4) applies. Split the source at the exit time $\tau=\tau_2$ (a stopping time, a.s. positive, by Lemma D22.1): pathwise,

\int_0^TS_t\dd t =\int_0^{T\wedge\tau}S_t\dd t+\one_{\{\tau<T\}}\int_\tau^TS_t\dd t \le\int_0^{T\wedge\tau}S_t\dd t+\one_{\{\tau\le T\}}\int_\tau^\infty S_t\dd t .

(D22.15)

Pre-exit term. By (I4) with $L=2$ ,

\E\int_0^{T\wedge\tau}S_t\dd t\ \le\ 8\cdot2^2\cdot T=32\,T .

(D22.16)

Post-exit term. Since $\{\tau\le T\}\in\mathcal F_\tau$ (Lemma D22.1(iv)) and $\{\tau\le T\}\subseteq\{\tau<\infty\}$ , the tower property and input (I5) at $\sigma=\tau$ give

\E\Bigl[\one_{\{\tau\le T\}}\int_\tau^\infty S_t\dd t\Bigr] =\E\Bigl[\one_{\{\tau\le T\}}\, \E\Bigl[\int_\tau^\infty S_t\dd t\Bigm|\mathcal F_\tau\Bigr]\Bigr] \le\E\Bigl[\frac{v_\tau}{\tau}\,\one_{\{\tau\le T\}}\Bigr].

(D22.17)

By Cauchy–Schwarz,

\E\Bigl[\frac{v_\tau}{\tau}\,\one_{\{\tau\le T\}}\Bigr] \le\Bigl(\E\bigl[v_\tau^2\,\one_{\{\tau\le T\}}\bigr]\Bigr)^{1/2} \Bigl(\E\bigl[\tau^{-2}\,\one_{\{\tau\le T\}}\bigr]\Bigr)^{1/2}.

(D22.18)

On the small-gap branch $\lambda\le3/(8K_n)$ , input (I6) gives $\E_\mu f^4\le2<\infty$ , so Lemma D22.3 with $E=\{\tau\le T\}$ yields

\E\bigl[v_\tau^2\,\one_{\{\tau\le T\}}\bigr]\ \le\ \E_\mu f^4\ \le\ 2 .

(D22.19)

Since $T\le T_0(n)\le\min\{1,t_1(n)\}$ , Lemma D22.4 applies, and combining (D22.17)–(D22.19),

\E\Bigl[\one_{\{\tau\le T\}}\int_\tau^\infty S_t\dd t\Bigr] \le\sqrt2\cdot\Bigl(5\bar C^2T^{-2}e^{-1/(\bar CT)}\Bigr)^{1/2} =\sqrt2\cdot\sqrt5\,\bar C\,T^{-1}e^{-1/(2\bar CT)} =\sqrt2\,T\cdot h(T),

(D22.20)

with $h$ as in Definition D22.1. Since $T\le t_c$ , $h(T)\le1$ , so

\E\Bigl[\one_{\{\tau\le T\}}\int_\tau^\infty S_t\dd t\Bigr]\ \le\ \sqrt2\,T .

(D22.21)

Adding (D22.16) and (D22.21),

\E\int_0^TS_t\dd t\ \le\ \bigl(32+\sqrt2\bigr)T\ \le\ 34\,T,

(D22.22)

which is (D22.13). Since $q(t)\ge0$ and $D_t\ge0$ , the form (21.11) with $C_0=34$ , $C_1=0$ follows a fortiori, with the entire damping budget untouched.

The fixed- $n$ bridge and the frontier reproduction¶

Proof D22.6

Fix $n\ge2$ and set $M_*=1+34\,t_c$ and

T_*(n)=\min\Bigl\{T_0(n),\ \frac1{2M_*}\Bigr\} =\min\Bigl\{t_c,\ \frac1{\bar C\log^2n},\ \frac1{2M_*}\Bigr\}.

(D22.24)

Step 1: the eigenvalue bound on regular isotropic approximants. Let $\mu$ be any regular isotropic approximant on $\R^n$ with first eigenvalue $\lambda$ .

Large-gap branch. If $\lambda>3/(8K_n)$ then directly

\CP(\mu)=\frac1\lambda<\frac{8K_n}3=\frac{8C_K}3\log n .

(D22.25)

Small-gap branch. If $\lambda\le3/(8K_n)$ , run the certified bridge argument of prop-spectral-sufficiency.md at this fixed $\mu$ , with Theorem D22.1 in place of the universal occupation hypothesis. The certified dossier’s $q$ -identity for the regular class,

q(t)=\abs{g_0}^2+\E\int_0^t\bigl(S_s-2D_s\bigr)\dd s,

(D22.26)

holds with all displayed integrals finite: $\E\int_0^tS\le34t$ by Theorem D22.1 for $t\le T_0(n)$ , while the certified posterior covariance cap $(\varepsilon+s)A_s\preceq I$ and the terminal cap $(\varepsilon+s)\abs{g_s}^2\le v_s$ give the crude ( $\varepsilon$ -dependent, finiteness-only) bounds $q(s)\le\varepsilon^{-1}$ and $\E D_s\le\varepsilon^{-2}$ . The certified Bessel bound gives $\abs{g_0}^2\le1$ . Dropping the nonnegative damping and applying Theorem D22.1,

q(t)\ \le\ 1+34\,t\ \le\ 1+34\,t_c=M_*, \qquad 0\le t\le T_0(n)

(D22.27)

(no Grönwall step is needed since $C_1=0$ ). The certified terminal-variance identity $\E\Var_{\mu_t}(f)=1-\int_0^tq(s)\dd s$ then gives, at $t=T_*(n)\le T_0(n)$ ,

\E\Var_{\mu_{T_*(n)}}(f)\ \ge\ 1-T_*(n)\,M_*\ \ge\ \frac12,

(D22.28)

while the certified posterior Brascamp–Lieb step (the posterior Hessian is $\succeq(\varepsilon+t)I\succeq tI$ ) with the fixed-test tower property bounds the same quantity above by $\lambda/T_*(n)$ . Hence

\lambda\ \ge\ \frac{T_*(n)}2 .

(D22.29)

Both branches. For $n\ge2$ one has $\log^2n\ge\log^22>0$ , so each of the three terms in $T_*(n)$ is bounded below by a universal multiple of $1/\log^2n$ :

T_*(n)\ \ge\ \frac{c_1}{\log^2n}, \qquad c_1:=\min\Bigl\{t_c\log^22,\ \frac{\log^22}{2M_*},\ \frac1{\bar C}\Bigr\},

(D22.30)

and on the large-gap branch $1/\lambda<\tfrac{8C_K}3\log n\le\bigl(\tfrac{8C_K}{3\log2}\bigr)\log^2n$ . Therefore every regular isotropic approximant on $\R^n$ satisfies

\CP(\mu)=\frac1\lambda\ \le\ C_2\,\log^2n, \qquad C_2:=\max\Bigl\{\frac{8C_K}{3\log2},\ \frac2{c_1}\Bigr\}\ \text{universal}.

(D22.31)

Step 2: passage to an arbitrary isotropic log-concave law at fixed $n$ . Let $\nu$ be isotropic log-concave on $\R^n$ . The certified bridge dossier constructs regular isotropic approximants $\nu_j\to\nu$ in $W_2$ , in the same dimension $n$ , each lying in the regular class with compact resolvent and a first nonconstant eigenfunction, and shows that a Poincaré inequality holding for every $\nu_j$ with a constant independent of $j$ passes to $\nu$ on fixed locally Lipschitz test functions (value truncation, Lipschitz cutoffs, mollification, and lower semicontinuity; no eigenfunction convergence is used). By (D22.31) the constant $C_2\log^2n$ is independent of $j$ (the dimension $n$ is fixed along the approximation), so exactly that certified passage yields

\Var_\nu(h)\le C_2\log^2n\int\abs{\nabla h}^2\dd\nu

(D22.32)

for every locally Lipschitz $h$ , i.e. $\CP(\nu)\le C_2\log^2n$ . Taking $C=C_2$ completes the proof.

Remark D22.2 (Where each dimension dependence enters)

Exactly two inputs carry the dimension: the window length $t_1(n)=1/(\bar C\log^2n)$ of the published import (I2), and the frontier constant $K_n=C_K\log n$ of the published import (I3) (which sets the branch point $3/(8K_n)$ and, through the small-gap branch, contributes only inside $T_*(n)$ via universal constants). All other constants ( $8L^2$ , 34, $\sqrt2$ , $M_*$ , $t_c$ ) are universal and $\varepsilon$ -free. The regularization $\varepsilon$ appears only in finiteness bounds and in the favorable denominator shift of the restart lemma; it enters no final constant. This uniformity concerns the displayed implications and the redundant Poincaré consequence; the small-gap implication still has an empty admissible class.

Obstructions respected. The node carries no bounded_by edge; the registered fences were checked one by one. rem:two-tail-slice-bounds: no cut, slice, or absolute-scale excess estimate occurs; the estimate is a stopped expectation bound plus a rare-event charge. rem:projection-ceiling: the only tensor input is the full symmetric-matrix Letwin bound, consumed inside the companion dossier in its certified all-law form; no projection test is promoted. rem:crude-insufficient: no crude covariance integral $\Xi_T$ and no logarithmic bootstrap appears; the window import is an exit-probability bound. rem:relative-ceiling: no all-measure relative occupation premise is inserted; the a priori input is the published frontier $K_n$ , and the output is strictly weaker than that input. rem:profile-circularity: no localized isoperimetric profile or moving competitor family occurs. rem:single-coordinate-cuts: no product-cut claim is made. Covariance spike (Proposition 0.1): respected constructively — it is the stated reason the mechanism stops at the window edge and no beyond-window claim is made. Recorded dead ends: the marginal-probability independence step is avoided (the post-exit charge is a joint Cauchy–Schwarz through Lemma D22.3, not a product of marginals); no unstopped $\norm{A_t}_\op$ moment, no moving projector, and no rank-tail entrance loss occurs. Trace-upgrade comparison constraint: only conj:mm-spectral-occupation is touched; no assertion crosses to the trace-upgrade cluster.

References¶

Letwin, B. (2026). The KLS Constant is O(\log1/4 n).
Klartag, B., & Lehec, J. (2025). Isoperimetric Inequalities in High-Dimensional Convex Sets. Bulletin of the American Mathematical Society, 62(4), 575–642. 10.1090/bull/1869
Klartag, B. (2023). Logarithmic Bounds for Isoperimetry and Slices of Convex Sets. Ars Inveniendi Analytica, 2023(4), 1–17. 10.15781/jsjy-0b06

Preliminary lemmas¶

The window occupation estimate¶

The fixed-nnn bridge and the frontier reproduction¶

The fixed- $n$ bridge and the frontier reproduction¶