The fixed cut: remaining problems - The KLS theorem and its methods

Part of the fixed-cut archive (Chapter The fixed cut: approach and lessons); this chapter holds the problems the fixed cut leaves, together with the obstructions that shaped them. None is needed for KLS, now proved; each is stated for the difficulty it isolates. The operator-to-trace upgrade below has a counterpart among the alternative mechanisms — the linear test of the moment map asks for an operator bound where only a trace bound is known (Section The linear test: the necessary linear-sector condition) — and the counterexamples of this chapter, Proposition 29.1 first, constrain any argument that follows a set through localization. We state the problems in decreasing order of strength.

By Corollary 30.1, this prefix form on the tight window is exactly what the argument uses; the stronger every-interval estimate Assumption 28.1 is not required. The projection-test ceiling of Chapter The static quadratic-chaos input and the two-tail obstruction (Remark 25.4) bears on the obvious proof method, although Letwin’s moment-map argument bypasses it for the intrinsic static quadratic chaos. A proof of the dynamic upgrade must therefore use cut-specific structure and covariance occupation. Its restriction to products is examined in Remark 27.3; a family of product cuts violating the prefix form for every choice of constants would refute it.

Proposition 29.1 (Exponential-spectator obstruction to global weighted propagation — Proved · agent review (gpt-6-astra, 2026-10-01) · prop:weighted-spectator-obstruction · Correction)

For every proposed constants $C,T_0,\gamma>0$ , every $\eta\in(0,1/2)$ , and every $\delta>0$ , there exist a dimension $d$ , an isotropic product $\mu$ of $d$ centered one-sided exponentials, a finite-perimeter cylinder $E$ with $\mu(E)=1/2$ ,

e_0(E)\le\delta, \qquad \frac{e_0(E)}{I_\mu(1/2)}\le\delta,

(29.3)

and a time $0<T\le T_0$ such that

\E\int_0^{T\wedge\tau_\eta} e_t(E)\bigl(1+\norm{A_t}_\op\bigr)^{5/2}\dd t >C\bigl(Te_0(E)+T^{1+\gamma}\bigr).

(29.4)

This proves that the global operator-norm weight is unstable under independent spectators and that the literal rate in Conjecture 29.2 fails. All witness measures are products, which satisfy KLS by Proposition 27.2.

Proposition 29.2 (Spectator obstruction to a superlinear excess remainder — Proved · agent review (2026-08-27) · prop:spectator-excess-rate-obstruction · Correction)

e_0(E)\le\delta, \qquad \frac{e_0(E)}{I_\mu(1/2)}\le\delta,

(29.5)

and a time $0<T\le T_0$ such that

\E\int_0^{T\wedge\tau_\eta}e_t(E)\dd t >C\bigl(Te_0(E)+T^{1+\gamma}\bigr).

(29.6)

Thus replacing the global operator-norm weight by a tensor-stable or cut-local weight is not by itself enough to repair the current propagation estimate: a surviving uniform statement must also allow an $O(T)$ remainder, make the remainder vanish with a source deficit, or impose an explicit near-worst-measure hypothesis. The product witnesses satisfy KLS by Proposition 27.2.

The propagation clause of the weighted package is the following rate. It is refuted by the product cylinders of Proposition 29.1 above, and stated here so that the refutation has a precise target.

Against this rate, Proposition 29.1 gives, for every proposed choice of constants, a product-cylinder witness with arbitrarily small additive and relative initial excess. Among pure powers of $\lmax(A_t)$ multiplying absolute excess in this slice-wise package, $5/2$ is the weakest exponent statically consistent with the two-tail mode (Proposition 25.2); the time exponent was a deliberately stronger demand, not fixed by that static example. The near-worst bootstrap of Theorem 33.1 supplies a different, externally anchored unweighted bound. It is not a uniform superlinear-remainder statement of the form above. Proposition 29.2 proves that the superlinear remainder already fails after the global covariance weight is removed. Thus changing only the weight cannot repair the uniform package: the replacement must also change the remainder or impose an explicit near-worst-measure hypothesis.

A possible replacement, screened by the source, charges the weighted excess only on the aligned set $\mathcal A_{\kappa,t}=\{Q_t\ge\kappa\,e_tW_{\rm cut}\}$ , where $Q_t=\calS_{\mu_t}(E)/s_t=s_t\norm{K_t}_{\HS}^2$ and $W_{\rm cut}=(1+\lambda_{\rm cut}(A_t,K_t))^{5/2}$ with the cut-oriented scale of Lemma 30.3. Its first positive result, Proposition 29.3 below, concerns the regular split class; general split laws are covered only through their regular approximants, because the interchange of the screened indicator with the approximation limit is not carried out.

Proposition 29.3 (Split-class screened supply, total-budget form — Proved · agent review (2026-08-30) · prop:split-screened-supply · Correction)

Let $\mu$ be a product of isotropic one-dimensional log-concave laws, each compactly supported with smooth density, and let $E=E_J\times\R^{J^c}$ for a fixed coordinate set $J$ with $\abs J=k$ , where $E_J\subset\R^J$ has $C^2$ relative boundary with a tubular neighborhood over the support, and $0<p_0<1$ . 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} \mathbf 1_{\{Q_t\ge\kappa e_tW_{\rm cut}\}}\dd t \ \le\ \frac{2k+64\eta^2(1+k)}{\kappa},

(29.8)

uniformly in the ambient dimension and in every spectator coordinate. The bound is a total budget, not of the form $C(k)\,T$ , and asserts nothing about the trace-upgrade cluster.

Conjecture 29.3 (Weighted stable Stein-trace estimate — Not settled here · conj:stein-weighted · Idea · Counterexample)

There exist universal constants $T_0,C_0,C_1,C_2>0$ , $0\le\beta<\tfrac12$ , and $\eta\in(0,\tfrac14]$ satisfying $2\beta+64\eta^2<1$ such that, for every isotropic log-concave law, every initial finite-perimeter cut $E$ with $e_0(E)\le1$ and $\abs{p_0-\tfrac12}\le\eta/2$ , and every $0<T\le T_0$ ,

\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}

(29.9)

This is exactly the stable Stein-trace clause of Assumption 28.3, considered as an independent analytic ingredient.

Taken alone, this clause neither repairs that package nor implies KLS: it needs a companion propagation statement that survives the spectator products. The intrinsic quadratic-chaos input is Theorem 25.1. The mechanism of Chapter The fixed cut: Reilly, Jacobi and splitting formulas needs in addition a localization-uniform quantitative almost-stability trace theorem for the fixed cut, modulo tangential Jacobi zero modes and with all Reilly boundary terms controlled (Conjecture 34.1). Proposition 25.2 rules out a slice-wise shortcut. No implication between this statement, Conjecture 29.1, and Conjecture 31.1 is asserted.

Remark 29.1 (A shared dynamic occupation problem, and a missing geometric bridge)

Conjecture 29.1 and Conjecture 29.3 (its high-rank, mean-zero part) and the adapted alignment problem Conjecture 31.1 all confront a high-rank occupation difficulty, but no equivalence between them is asserted. Analytically, the gap of the all-cut variant is exactly the trace of the occupation operator $M:=\E\int_0^\infty s_t G_t^2\dd t$ : Corollary 24.1 is the statement $M\preceq R_0\preceq I$ (the diagonal/quadratic-form level), and the trace target is $\Tr(M)$ , whose naive bound $\sum_i(R_0)_{ii}=\Tr R_0\le n$ is the obstruction. Geometrically, the proposed Reilly/Jacobi approach aims to control analogous high-rank boundary modes, but Conjecture 34.1 records the missing trace bridge; its proposed constant-mode branch is Conjecture 29.5. Combinatorially, summing the per-coordinate budgets of Theorem 31.1 over a product cut is again $\Tr R_0\le n$ ; this motivates the incident-high residue in Conjecture 31.1 but does not identify it with the full trace problem. The product budget argument supplies only the naive trace bound of order $n$ , while the dimension-dependent early-window estimate follows separately from covariance moments (Theorem 31.3, at the $c/\log n$ scale). This motivates — but does not prove — an occupation-density bound: the source-occupation measure on time $\,\times\,$ direction should have bounded density on the early balanced window, so that only $O(1)$ directions are simultaneously active. This is the form in which Conjecture 31.1 should be integrated against time.

Conjecture 29.4 (Extremality tames the covariance process — Not settled here · conj:taming · Idea · Counterexample)

For every $\kappa\in(0,1]$ there exist $\eps>0$ and $T_0\in(0,1/8)$ such that every isotropic log-concave $\mu$ on $\R^n$ with $h_\mu\le(1+\eps)\hstar_n$ satisfies $h_\mu\bigl(T_0^{4/3}+\Xi_{T_0}(\mu)\bigr)\le\kappa\,T_0$ . By Theorem 33.1 this gives the absolute-scale supply $T_0e_0+C\kappa T_0$ for near-worst measures. At a sufficiently small universal time, the stronger, unweighted condition $\Xi_{T_0}\le\kappa T_0$ for every measure is already KLS-sufficient by Proposition 33.1; it is not the statement conjectured here. Heuristically, covariance inflation under localization is driven by third-moment anisotropy, while the splitting philosophy of Chapter The fixed cut: Reilly, Jacobi and splitting formulas predicts that near-worst measures are approximately split along their dangerous directions, where the variance processes are one-dimensional and tame (Proposition 27.2(ii)); making this rigorous would couple the splitting analysis to the covariance SDE.

Remark 29.2 (The naive splitting–covariance coupling faces a type mismatch)

The coupling suggested above runs into a structural obstruction: $\Xi_T$ is a cut-free functional of the covariance process, whereas the splitting structure is cut-indexed. Product/splitting alone does not suppress coordinatewise covariance excursions; in the one-sided-exponential product stress test, a split coordinate can also contribute to $\lmax(A_t)$ . Thus the splitting philosophy controls the source of the cut-aware Stein estimate (Conjecture 29.3, Conjecture 29.5), not the cut-free $\Xi_T$ . The residual non-circular content of Conjecture 29.4 is therefore a near-worst-specific improvement of the Klartag–Lehec window precisely in the regime $\hstar_n\gg1/\log\log n$ that Corollary 33.1 does not reach.

Assumption 29.1 (Absolute-scale geometric completion — Not settled here · ass:absolute-geometric-completion · Idea · Counterexample)

There exist universal constants $T_0\in(0,1/8)$ and $\kappa,c_g,\eps_g,\delta_g>0$ with the following property. Set $\eta=T_0^{1/3}$ . For every integer $n\ge2$ , every isotropic log-concave probability measure $\mu$ on $\R^n$ satisfying $h_\mu\le(1+\eps_g)\hstar_n$ , and every measurable cut $E$ with $\mu(E)=1/2$ and

0\le e_0(E)=\mu^+(E)-I_\mu(1/2)\le\delta_g,

(29.10)

the supply estimate

\int_0^{T_0}\E\bigl[\bar e_t(E)\one_{\{t<\tau_\eta\}}\bigr]\dd t \le T_0e_0(E)+\kappa T_0

(29.11)

implies $\mu^+(E)\ge c_g$ .

The bound of Corollary 29.1 is conditional on the geometric completion. Before KLS was proved, it improved on Theorem 4.6: it would replace $\PsiKLS_n\lesssim(\log n)^{1/4}$ by $\PsiKLS_n\lesssim1+\log\log n$ in sufficiently large dimensions, and so $C_{\mathrm P,n}\lesssim\sqrt{\log n}$ by $C_{\mathrm P,n}\lesssim(1+\log\log n)^2$ , still growing with dimension. KLS, now proved (Chapter The proofs of KLS compared), gives $\PsiKLS_n=O(1)$ and supersedes both. The corollary is kept for its mechanism: the completion supplies the geometric step converting a small accumulated excess into a perimeter lower bound, and the only covariance input is the published window of Corollary 26.2, not the longer $c/\log n$ window.

Proof 29.1

Fix the completion constants and use exactly $\eta=T_0^{1/3}$ , with $\eps=\tfrac12\min\{1,\eps_g,T_0^{1/3}\}$ . Choose $N\ge3$ so that the covariance cutoff $t_1(n)=c_0(\log n)^{-2}$ from Corollary 26.2 is at most $T_0$ and $L_n:=1+\log\log n\ge1$ for all $n\ge N$ . If $C_L$ is the constant in (33.11), set $a=\min\{\kappa T_0/(4C_L),c_g/2\}$ . Suppose $\hstar_n\le a/L_n$ . Its positivity (Theorem 26.2) and finiteness allow a near-worst measure with $h_\mu\le(1+\eps)\hstar_n\le2\hstar_n$ . Choose balanced cuts $E_j$ whose excesses satisfy $0\le e_j<\min\{\delta_g,j^{-1}\}$ ; this uses the infimum defining $I_\mu(1/2)$ , not an exactly minimizing cut. For each fixed cut and its own stopping time, Corollary 33.1 gives

\int_0^{T_0}\E[\bar e_t(E_j)\one_{\{t<\tau_\eta(E_j)\}}]\dd t \le T_0e_j+C_Lh_\mu(T_0^{4/3}+L_n) \le T_0e_j+4C_L\hstar_nL_n \le T_0e_j+\kappa T_0.

(29.13)

The completion therefore gives $\mu^+(E_j)\ge c_g$ for every $j$ . Letting only the scalar perimeter values tend to $I_\mu(1/2)=h_\mu/2$ yields $h_\mu\ge2c_g$ , whereas $h_\mu\le2a/L_n\le c_g$ , a contradiction. Finally $L_n\le2\log\log n$ for sufficiently large $n$ .

A dimension-free conclusion needs a further estimate at the completion’s same time $T_0$ . More precisely, let $C_B$ be the constant in (33.7). If some universal $\eps_t>0$ ensures

h_\mu\bigl(T_0^{4/3}+\Xi_{T_0}(\mu)\bigr) \le\frac{\kappa}{C_B}T_0

(29.14)

for every $n\ge2$ and every isotropic log-concave $\mu$ with $h_\mu\le(1+\eps_t)\hstar_n$ , then the clean bootstrap supplies the completion’s hypothesis. The balanced near-minimizer argument above gives $h_\mu\ge2c_g$ for arbitrarily accurate near-worst measures; taking their near-worst error to zero gives $\hstar_n\ge2c_g$ . Monotonicity in Lemma 33.2 covers dimension one, yielding the Cheeger formulation of Conjecture 0.1. In Conjecture 29.4, however, the time is chosen after the requested error level. Its existential quantifier does not identify that time with the completion time. The extra estimate displayed here is thus a sufficient condition beyond the implication in Corollary 29.1.

An effective rank in place of the soft maximum¶

In the localization–Lichnerowicz argument, Remark 2.2 attributes the $\log n$ cost to the fact that the soft maximum (2.11) approximates $\lmax$ over $n$ directions. A potential depending only on the directions actually relevant to a near-extremizer — or on an effective rank rather than the ambient dimension — could convert $\kappa_n=O(1)$ into $\CP=O(1)$ , if the entropy cost of the soft maximum were the only loss. The quantity $\Xi_T(\mu)$ of (28.8), evaluated in Chapter The fixed cut: the bootstrap, is the fixed cut’s version of this question, and Conjecture 29.4 is the corresponding statement. Chapter The fixed cut: the bootstrap explains why the crude evaluation cannot suffice (Remark 33.2, Remark 33.5) and shows that a relative bound at a sufficiently small universal time would itself give KLS (Proposition 33.1).

References¶

Bobkov, S. G., & Chistyakov, G. P. (2015). On Concentration Functions of Random Variables. Journal of Theoretical Probability, 28(3), 976–988. 10.1007/s10959-013-0504-1