The fixed cut: approach and lessons - The KLS theorem and its methods

This chapter opens the fixed-cut archive, formed by this chapter and the six that follow. The fixed cut was the first localization argument of this manuscript. KLS is now proved by other means (Chapter The proofs of KLS compared), and the fixed cut is kept for its results, which are obstructions, a ceiling and counterexamples, and which apply beyond it. The ceiling Proposition 33.1 measures how strong one covariance input would have to be for a bootstrap of the boundary excess to close. The spectator products of Proposition 29.1 refute a natural weighted propagation estimate, and with it any global covariance weight that charges coordinates a cut does not use. The product stress test of Chapter The fixed cut: product stress test is the tensorization test of Section The tensorization test carried out on an all-cut estimate. Any argument that follows a set through localization meets all three. The open statements of the archive are precise problems, not steps towards KLS.

The all-cut and near-Cheeger variants¶

Both variants follow a fixed cut under the same stochastic localization; they differ in which cuts they must control, and the two names used throughout the archive come from this difference.

The all-cut variant asks for an absorptive two-color Carleson estimate for every balanced cut (Assumption 28.1 below); this alone yields KLS (Theorem 28.1).

The near-Cheeger variant works only with near-minimizers of the isoperimetric profile, the cuts that a proof of KLS by contradiction has to handle. It couples weighted excess propagation under localization to a weighted stable Stein-trace estimate (Assumption 28.3 below). Its literal form fails on the product cylinders described under What rules out the obvious variants below. A workable replacement must use a cut-local or tensor-stable covariance scale and also permit an $O(T)$ supply, use a genuinely source-tied remainder, or impose an explicit near-worst-measure hypothesis.

The near-Cheeger variant is the natural home for the boundary and Jacobi ideas of Chapter The fixed cut: Reilly, Jacobi and splitting formulas. It gives no all-cut estimate: arbitrary balanced sets can have enormous two-color covariance contrast even when their perimeter is small in an anisotropic posterior.

The approach at a glance¶

The summary below has the format of those of the three alternative mechanisms, compared in Chapter Alternative mechanisms after KLS.

The idea. Follow one fixed would-be bottleneck cut $E$ under Eldan’s stochastic localization, and show that it cannot be identified too quickly.

What it would add. The fixed cut is kept as the record of a method: its obstructions and its ceiling are its results, and they say what any argument that follows a single cut through localization has to overcome. Were its hypotheses established, it would give KLS through a chain of two implications, conditional at exactly one place, through a bottleneck set that cannot be identified too quickly:

\text{a balanced cut survives to a universal time} \ \Longrightarrow\ \text{boundary lower bound} \ \Longrightarrow\ \text{KLS}.

(28.1)

The survival-to-boundary step is Lemma 30.1. To obtain survival, Theorem 30.2 converts the all-cut Carleson hypothesis into stopped centroid control, and Theorem 30.1 uses the mass martingale. The weaker hypothesis on deterministic prefixes, Assumption 28.2, is what Corollary 30.1 uses; Theorem 28.1 records the consequence of the stronger all-cut formulation. The near-Cheeger implication is Theorem 28.2; the product witnesses of Proposition 29.1 refute its propagation hypothesis. None of these implications supplies its own Carleson or centroid hypothesis.

What it uses. From the literature: the localization process and its covariance SDE, and the improved Lichnerowicz estimate (Theorem 3.1). Developed in this manuscript: the two-color Riccati identities (Theorem 24.1, Chapter The two-color Riccati identities), the Stein dictionary (Section The Stein dictionary), and the small-time covariance control (Chapter Small-time operator-norm control of the covariance). Each statement below displays its own standing.

What it gives. Its implications towards KLS are conditional, and its results are its obstructions and its ceiling; this chapter states both, and Chapters The fixed cut: the mass martingale and the Carleson estimate to The fixed cut: Reilly, Jacobi and splitting formulas hold the arguments.

A bootstrap, and the one quantity it needs. Theorem 33.1 says that for a measure whose Cheeger constant is within a factor $1+\varepsilon$ of the smallest in its dimension, the excess of every balanced cut propagates along localization with a single quantity, $\Xi_T(\mu)=\int_0^T\E(\lmax(A_t)-1)_+\dd t$ . The idea: the isoperimetric profile of a log-concave measure is concave and symmetric, so its Cheeger constant is read at mass $\tfrac12$ ; whitening a localized measure of covariance $A_t$ loses at most $\lmax(A_t)^{1/2}$ against the worst isotropic measure of the same dimension; near-worstness compares the localized Cheeger constant with that of $\mu$ , and every loss is charged to $(\lmax(A_t)-1)_+$ .
How far it gets. The crude evaluation $\Xi_T\lesssim\log n$ never suffices in high dimension (Remark 33.2, Remark 33.5); the polylogarithmic covariance estimates improve it to $\Xi_T\le C(1+\log\log n)$ for $t_1(n)\le T\le1$ (Corollary 33.1), which still falls short of the $O(T)$ the bootstrap needs. A relative bound $\Xi_{T_0}(\mu)\le\kappa T_0$ for all measures at a sufficiently small universal time would itself give KLS (Proposition 33.1): this measures the input the bootstrap needs, without excluding other propagation arguments.
Budgets. The scale-weighted all-cut source budget Lemma 29.1, and the coordinate budgets and covariance reduction of the product stress test (Chapter The fixed cut: product stress test), where the approach meets the measure that defeats naive covariance control: no fixed balanced cut of a product that depends on a single coordinate can witness a failure of the all-cut chain of implications, since its conclusion holds for such cuts unconditionally (Corollary 31.1).
The excess term is inert. Unweighted excess propagation holds unconditionally at scale $O(T)$ , so the unweighted Stein-trace estimate alone would imply KLS, and that estimate admits no proof one time-slice at a time (Proposition 28.1). The whole weight of the near-Cheeger variant is carried by the trace estimate; a covariance weight is what makes the excess term matter.

What blocks it. For the all-cut variant, the operator-to-trace upgrade Conjecture 29.1: obtain Assumption 28.2 by lifting Corollary 24.1 from quadratic-form scale to trace scale, uniformly over fixed initial data (Chapter The fixed cut: remaining problems). For the near-Cheeger variant, the trace estimate Conjecture 29.3 lacks a companion: a tensor-stable propagation statement to replace the refuted literal package Assumption 28.3, together with a localization-uniform almost-stability trace theorem (Conjecture 34.1). For the bootstrap, the missing input is a near-worst bound on $h_\mu\,\Xi_T$ at a universal time (Conjecture 29.4); the constant-mode branch of the Reilly–Jacobi mechanism rests on quantitative splitting (Conjecture 29.5).

What rules out the obvious variants. Two obstructions, which are the most reusable output of this approach. A global operator-norm covariance weight, as in Assumption 28.3 and Conjecture 29.2, charges independent spectator coordinates that contribute nothing, and a balanced cylinder in a product of one-sided exponentials makes the overcharge unbounded (Proposition 29.1). The same spectators rule out a uniform superlinear remainder even with weight one (Proposition 29.2). Separately, a single inflated coordinate cannot carry a counterexample (Corollary 31.1, Section Coordinate budgets, and the single-coordinate two-tail scenario). Two further constraints are methodological warnings rather than theorems: the two-tail obstruction (Remark 25.3) and the circularity warning of Section Why direct propagation risks circularity.

What would settle it. The all-cut variant is settled positively by Conjecture 29.1 with universal constants, and negatively by a family of balanced cuts on which the trace-scale statement fails. The near-Cheeger variant first needs a new formulation: a cut-local, tensor-stable covariance weight that ignores independent spectators while still dominating the aligned two-tail mode, or a replacement carrying an explicit near-worst-measure hypothesis.

Where to read. Conceptual prelude: Chapter Prelude: what stochastic localization does, and what it costs. The static quadratic-chaos input and its limits: Chapter The static quadratic-chaos input and the two-tail obstruction. The fixed cut itself: this chapter for the conditional statements and what they teach, Chapter The fixed cut: remaining problems for the remaining problems, and then the arguments — the mass martingale and the all-cut implication (Chapter The fixed cut: the mass martingale and the Carleson estimate), the product stress test with its calculations (Chapter The fixed cut: product stress test), the near-Cheeger variant with the Stein dictionary and excess propagation (Chapter The fixed cut: the near-Cheeger variant), the bootstrap (Chapter The fixed cut: the bootstrap), and the Reilly, Jacobi and splitting formulas (Chapter The fixed cut: Reilly, Jacobi and splitting formulas). Shared apparatus: Chapters Analytic conventions and the two-color localization setup, The two-color Riccati identities, The static quadratic-chaos input and the two-tail obstruction, Small-time operator-norm control of the covariance and Model geometries: the Gaussian and product brackets. Full proofs are linked from the status shown next to each statement.

The idea, in one line.

\boxed{\text{follow the posterior evolution of a fixed would-be bottleneck cut }E.}

(28.2)

The mass $p_t=\mu_t(E)$ is a martingale. KLS follows if a balanced near-bottleneck cut cannot be identified too rapidly by the early Gaussian observation generated by localization. Retaining one set, rather than the full spectrum of $A_t$ , is this approach’s response to Proposition 0.1.

Throughout, $\mu$ is isotropic log-concave on $\R^n$ , so $\E_\mu X=0$ and $\Cov_\mu(X)=I_n$ ; the isoperimetric profile is

I_\mu(p)=\inf\{\mu^+(E):\mu(E)=p\},\qquad 0<p<1,

(28.3)

and $h_\mu$ , $\PsiKLS_\mu=h_\mu^{-1}$ , $\hstar_n$ are as fixed in Section The conjecture and (0.5); recall in particular The $\psi$ convention on the $\psi$ convention. KLS and the state of the literature are in Sections The question and What was known; the sharp radial and homogeneous-quadratic results recorded there remove major static obstructions, but they do not by themselves control a fixed bottleneck set under localization.

Main stochastic quantities¶

For orientation we recall the two-color quantities; they are defined in full in Chapter Analytic conventions and the two-color localization setup. Fix a measurable set $E$ and put $F=E^c$ . Under localization define

p_t=\mu_t(E),\qquad q_t=1-p_t, \qquad s_t=p_tq_t,

(28.4)

\delta_t=m_t^E-m_t^F, \qquad G_t=\Sigma_t^E-\Sigma_t^F, \qquad r_t=s_t\abs{\delta_t}^2,

(28.5)

where $m_t^E,m_t^F$ and $\Sigma_t^E,\Sigma_t^F$ are the two conditional means and covariances. The Riccati source and damping are

S_t=s_t\norm{G_t}_{\HS}^2, \qquad D_t=2s_t\delta_t^TA_t\delta_t-r_t^2,

(28.6)

where $A_t=\Cov(\mu_t)$ . The isoperimetric excess process is

e_t(E)=\mu_t^+(E)-I_{\mu_t}(p_t)\ge0,

(28.7)

and the covariance-excess functionals, central to both variants, are

X_t=\bigl(\lmax(A_t)-1\bigr)_+, \qquad \Xi_T(\mu)=\int_0^T\E X_t\dd t .

(28.8)

Let $\tau$ be the coarse balanced exit time (23.18), the first time at which $p_t$ leaves $[1/3,2/3]$ , and let $\tau_\eta$ be the tight window of (23.19) below; balanced initial cuts are as in Chapter Analytic conventions and the two-color localization setup, $p_0\in[2/5,3/5]$ for the coarse window.

Main conditional statements¶

The argument is in Chapter The fixed cut: the mass martingale and the Carleson estimate. Taking $I=[0,T]$ and $\eta=1/6$ , for which the tight window is the coarse one, the assumption gives the hypothesis of Corollary 30.1 for every cut of mass $1/2$ ; Lemma 30.1 then concludes. It uses only stochastic localization, the two-color Riccati identity, and the fact that a $T$ -uniformly log-concave posterior has a dimension-free Cheeger lower bound at scale $\sqrt T$ .

The step from a Carleson estimate to a boundary lower bound, Corollary 30.1, needs less than the all-interval formulation above. Its exact hypothesis is the following.

This prefix-only statement is implied by Assumption 28.1: take $I=[0,T]$ and $\eta=1/6$ , for which (23.19) is (23.18); no converse is known. Corollary 30.1 derives its boundary-lower-bound consequence for each cut; universal validity would therefore give KLS by the same balanced near-minimizer reduction.

The hypothesis of the conditional theorem Theorem 28.2 of the near-Cheeger variant is the following package.

Assumption 28.3 (Literal weighted near-Cheeger package — Refuted by Proposition 29.1 · ass:weighted-package · Correction)

There are universal constants $T_0>0$ , $C_0,C_1,C_2\ge0$ , $0\le\beta<\tfrac12$ , $\gamma>0$ , and $\eta\in(0,\tfrac14]$ satisfying

2\beta+64\eta^2<1,

(28.11)

such that, for the corresponding fixed tight stopping window $\tau_\eta$ , the following two estimates hold for every initial finite-perimeter set $E$ with $e_0(E)\le1$ and $p_0=1/2$ (or $\abs{p_0-1/2}\le\eta/2$ ).

(i-w) Weighted excess propagation.

\E\int_0^{T\wedge\tau_\eta} e_t(E)\,\bigl(1+\norm{A_t}_\op\bigr)^{5/2}\dd t \le C_2\bigl(T e_0(E)+T^{1+\gamma}\bigr), \qquad 0<T\le T_0 .

(28.12)

(ii-w) Weighted stable Stein-trace estimate. With the Stein-trace norm $\calS_{\mu_t}(E)$ defined in Chapter The fixed cut: the near-Cheeger variant,

\begin{aligned} \E\int_0^{T\wedge\tau_\eta}\frac{\calS_{\mu_t}(E)}{s_t}\dd t &\le C_0T+C_1\E\int_0^{T\wedge\tau_\eta}r_t\dd t +\beta\E\int_0^{T\wedge\tau_\eta}D_t\dd t \\ &\qquad +C_2\E\int_0^{T\wedge\tau_\eta} e_t(E)\bigl(1+\norm{A_t}_\op\bigr)^{5/2}\dd t . \end{aligned}

(28.13)

Proposition 29.1 exhibits, for every choice of the constants, a product cylinder on which item (i-w) fails with these global quantifiers, even when the initial cut has arbitrarily small additive and relative excess. The package is therefore refuted; it is kept because it is the hypothesis of the next theorem, which records what it would have given.

The argument is in Section From the weighted package to KLS, using the excess bounds of Section Excess propagation; it is a conditional argument, valid whether or not its hypothesis holds. The weight $(1+\norm{A_t}_\op)^{5/2}$ is calibrated on the anisotropic two-tail configuration (Proposition 25.2): among pure powers of $\lmax(A_t)$ multiplying absolute excess in this slice-wise package, $5/2$ is the smallest statically consistent exponent. Static consistency is not enough: the same global norm also sees irrelevant independent spectators. The following proposition explains the first constraint, and Proposition 29.1 supplies the second.

Proposition 28.1 (Excess audit; deceptive strength of the unweighted estimate — Proved · agent review (2026-08-27) · prop:intro-audit · Correction)

(a) For every isotropic log-concave $\mu$ , every Borel set $E$ with $\mu(E)=1/2$ , finite initial lower outer Minkowski perimeter, and excess $e_0=e_0(E)$ , every $T>0$ and every stopping time $\tau$ ,

\E\int_0^{T\wedge\tau}e_t(E)\dd t\ \le\ (1+e_0)\,T .

(28.14)

(b) Consequently, the unweighted stable Stein-trace estimate — estimate (28.13) with the weight $(1+\norm{A_t}_\op)^{5/2}$ replaced by 1 — implies KLS by itself; the unweighted excess term is inert, and unweighted excess propagation is unconditionally true in the form used.

(c) Nevertheless, the unweighted estimate admits no slice-wise proof: there are log-concave pairs $(\nu,E)$ with $\nu(E)=1/2$ , $r=D=0$ , excess $e(E)\to0$ , and $\calS_\nu(E)/s\to\infty$ (Proposition 25.2).

The argument is in Section Excess propagation. Item (b) is not good news about excess propagation; it is a warning that the unweighted geometric package concentrated its entire logical weight, invisibly, in the Stein-trace estimate. Item (c) shows that this weight cannot be discharged one time-slice at a time. The proposed weighted formulation made the excess term non-inert and passed the static two-tail calibration, but the spectator obstruction shows that its global-operator-norm version is not tensor-stable. The separate unweighted spectator obstruction also rules out every uniform superlinear source-vanishing remainder of the same form. A replacement formulation would have to meet the static, tensor-stability and time-scale constraints together; changing the covariance weight alone does not suffice.

Finally, Chapter The fixed cut: the bootstrap provides the available propagation mechanism: a bootstrap comparison theorem anchored at $\hstar_n$ , valid for every balanced cut of a near-worst measure — the only regime a proof of KLS by contradiction requires — which compresses excess propagation into the scalar quantity $h_\mu\,\Xi_T(\mu)$ , together with a complete evaluation of that quantity against the known covariance estimates.