The fixed cut: product stress test

Part of the fixed-cut archive (Chapter The fixed cut: approach and lessons); this chapter tests the all-cut estimate on product measures, where KLS is known, and ends with the calculations behind its theorems (Section Calculations).

This chapter carries out Remark 27.3: it stress-tests the all-cut absorptive Carleson estimate (Assumption 28.1) on product measures, where (i) localization preserves product structure, (ii) KLS is known (Proposition 27.2), and (iii) the operator norm of $A_t$ genuinely reaches $\log n$ , so a proof cannot go through $\lmax$ -control. What a failure would decide is said in Remark 27.3: it concerns the every-interval form, reaches the prefix form Assumption 28.2 only if the same cuts violate it, and does not select the near-Cheeger variant, whose literal package the product witnesses of Proposition 29.1 already violate.

Three core tools are pointed at the product model. First, the per-direction Carleson estimate (Corollary 24.1) is a budget: $\E\int_0^\infty s_t\abs{G_t\theta}^2\dd t\le\theta^TR_0\theta\le1$ for every fixed direction. Second, the Stein identity $\calS_\nu(E)=s^2\norm K_\HS^2$ (Proposition 32.1) makes the quadratic-chaos machinery of Chapter The static quadratic-chaos input and the two-tail obstruction directly applicable to the Riccati source. Third, the small-time covariance control of Chapter Small-time operator-norm control of the covariance supplies the cut-free comparison quantities. Notation is that of this manuscript throughout: $\mu_t$ is Eldan stochastic localization Eldan, 2013Lee & Vempala, 2024, $p_t,s_t,\delta_t,G_t,K_t,r_t,S_t,D_t,R_t$ are the two-color quantities, $\tau$ is the coarse balanced exit time, and $X_t=(\lmax(A_t)-1)_+$ .

We also record the structural facts about products from Proposition 27.2: if $\mu=\bigotimes_{i=1}^n\mu^{(i)}$ with isotropic one-dimensional log-concave factors, then $\mu_t$ is a product pathwise (the tilt factorizes), $A_t=\diag(A_t^{(1)},\dots,A_t^{(n)})$ with each $A^{(i)}$ a one-dimensional variance process of drift $-(A^{(i)})^2$ , and $h_{\mu_t}\ge c\,\lmax(A_t)^{-1/2}$ pathwise, so that KLS for products is not in question; the question is whether the estimate under test holds there.

Coordinate budgets, and the single-coordinate two-tail scenario¶

Proof 31.1

For $i\notin J$ , the coordinate $x_i$ is independent of $\one_E$ , jointly with each $x_j$ . Hence $\E[x_i\mid E]=\E[x_i]$ , giving $\delta_i=0$ ; for $j\in J$ , $\Cov(x_i,x_j\mid E)=\E[x_i]\,\E[x_j\mid E]-\E[x_i]\,\E[x_j\mid E]=0$ , using $\E[x_ix_j\one_E]=\E[x_i]\E[x_j\one_E]$ ; and for $j\notin J$ , $j\ne i$ , $\Cov(x_i,x_j\mid E)=0=\Cov(x_i,x_j)$ , while for $j=i$ the conditional variance is unchanged. The same holds for $F=E^c$ , so all entries of $G=\Sigma^E-\Sigma^F$ outside the $J\times J$ block vanish, and likewise for $K=G+(q-p)\delta\delta^T$ .

Theorem 31.1 (Coordinate-budget theorem — Proved · agent review (2026-08-27) · thm:budget · Correction)

Let $\mu=\bigotimes_i\mu^{(i)}$ be a product of isotropic one-dimensional log-concave measures, and let $E$ be measurable with respect to $k$ coordinates, with $p_0\in[2/5,3/5]$ . Then:

(i) (Total source budget) $\displaystyle \E\int_0^\infty S_t\dd t\ \le\ \sum_{i\in J}(R_0)_{ii}\ \le\ k$ ;

(ii) (Bounded information rate) $\E\,r_{t\wedge\tau}\le1+k$ for all $t$ ;

(iii) (Boundary bound) there is a universal $c>0$ with

\mu^+(E)\ \ge\ \frac{c}{\sqrt{1+k}}\;\min(p_0,q_0).

(31.1)

Proof. The pathwise product structure confines $G_t$ to the $J\times J$ block, so the source splits coordinatewise and the per-direction Carleson estimate (Corollary 24.1) can be applied one direction at a time and summed. The calculation is carried out in Section Calculations.

Corollary 31.1 (Fixed single-coordinate cuts: bounded source and self-extinguishing two-tail spikes — Proved · agent review (2026-08-27) · cor:single-coordinate-cuts · Correction)

Let $\mu$ be a product as above and let $E$ be any fixed cut with $p_0\in[2/5,3/5]$ depending on one fixed coordinate $i$ , chosen before localization. This includes a fixed two-tail cut in a coordinate whose variance subsequently inflates, but not a pathwise choice of $E$ or $i$ . Then

\E\int_0^\infty S_t\dd t\ \le\ 1 \qquad\text{and}\qquad \mu^+(E)\ \ge\ c\,\min(p_0,q_0),

(31.2)

with universal $c$ . Consequently no such cut can witness a failure of the chain of implications from Assumption 28.1: the conclusion that the fixed-cut approach derives from the Carleson estimate holds for these cuts unconditionally. Quantitatively, the source spends only a vanishing fraction of time at large height: for every deterministic level $L>0$ ,

\E\,\bigl|\{t\ge0:\ S_t\ge\tfrac12L^2\}\bigr| \ \le\ \frac2{L^2}\,\E\int_0^\infty S_t\dd t\ \le\ \frac2{L^2}.

(31.3)

Thus a two-tail configuration of source height $S_t\asymp\Lambda^2$ (Proposition 25.2 with $\Lambda=A_t^{(i)}$ ), wherever along the path the coordinate variance $A_t^{(i)}$ inflates to order $\Lambda$ , is self-extinguishing: a spike to height of order $\Lambda^2$ is paid at that rate from a per-coordinate budget of total size 1, so the time spent above any fixed fraction of it is $O(\Lambda^{-2})$ .

Remark 31.1 (What Corollary 31.1 does and does not say)

Corollary 31.1 verifies, for single-coordinate cuts, the total stopped source bound — which is what the Riccati–Gronwall argument of Theorem 30.2 actually uses — not the literal interval form of Assumption 28.1; the distinction is immaterial for the argument, since that argument needs only $\E r_{t\wedge\tau}$ bounded, which (ii) provides directly. The corollary rules out the natural attack in which the cut and the inflating coordinate are fixed before localization. It says nothing about selecting the coordinate or cut after observing the path. The per-direction estimate is exactly the tool that disposes of the fixed-coordinate version. The point of the test — a proof of Assumption 28.1 cannot go through $\lmax(A_t)$ — is thus complemented on the counterexample side: a counterexample cannot go through a single inflated coordinate either. Both a proof and a counterexample are forced into genuinely high-dimensional, cut-specific territory; Theorem 31.1 localizes the entire remaining danger of the product model in cuts of unbounded coordinate complexity.

The covariance reduction and its general quadratic-chaos input¶

For cuts of unbounded complexity the block structure is unavailable. Product structure gives an elementary pathwise quadratic-chaos bound at every time. For general log-concave posteriors, the corresponding input is Theorem 25.1, used through Corollary 25.1. The general-measure implications below retain that explicit antecedent; the unwhitening loss remains the cut-free covariance factor.

Proof. Expand the quadratic form and use independence and centering to kill every cross-covariance, leaving a diagonal and an off-diagonal sum. The calculation is carried out in Section Calculations.

Lemma 31.2 remains a preprint-independent proof of the input on product paths. For general log-concave posteriors it is replaced by Corollary 25.1.

Corollary 31.2 (Covariance reduction of the all-cut estimate — Proved · agent review (gpt-6-astra, 2026-10-01) · cor:V2-implies · Correction)

Define the second-moment interface condition on a window $[0,T_0]$ :

\tag{V2} \int_I\E X_t^2\dd t\ \le\ K\,\abs I \qquad\text{for every interval }I\subset[0,T_0].

(31.6)

Conditional on Theorem 25.1, if (V2) holds for the localization of an isotropic log-concave measure $\mu$ , then the all-cut estimate (28.9) holds for $\mu$ on $[0,T_0]$ with $C_0=C(1+K)$ , $C_1=0$ , and damping coefficient $\alpha=0$ . For product measures the implication is unconditional. In that case it reduces the estimate to a statement about $n$ independent one-dimensional variance processes, with no reference to cuts.

The second-moment window and the sharpness regime¶

The second-moment bound through the $1/\log n$ window¶

The required covariance input is the fixed-time operator-norm moment control of Chapter Small-time operator-norm control of the covariance. The published sup-over-time estimate Theorem 26.1 supplies a preprint-independent $c/\log^2n$ fallback. Letwin’s quadratic-chaos input and the Klartag–Lehec moment theorem together extend the fixed-time second-moment window to $c/\log n$ .

Theorem 31.3 (Letwin-enhanced second-moment interface window — Proved · agent review (gpt-6-astra, 2026-10-01) · thm:V2-window · Correction)

Conditional on Theorem 25.1, there are universal constants $c_0,K_0>0$ such that for every isotropic log-concave $\mu$ on $\R^n$ , $n\ge2$ , and every $0\le t\le t_1(n):=c_0(\log n)^{-1}$ ,

\E\,X_t^2\ \le\ K_0 , \qquad\text{hence \textup{(V2)} holds on }[0,t_1(n)]\text{ with }K=K_0 .

(31.7)

Consequently, conditional on the preprint, Corollary 31.2 gives the inequality (28.9), measure by measure, for every initial log-concave $\mu$ (including products) on this dimension-dependent window. For products the implication (V2) $\Rightarrow$ (28.9) is preprint-independent, but the present $c/\log n$ verification of (V2) still uses the preprint. None of this establishes Assumption 28.1, whose time window must be universal in $n$ . Without the preprint, Theorem 26.1 verifies (V2) for every measure on $[0,c_0(\log n)^{-2}]$ ; the all-cut conclusion then follows preprint-independently for products via the product case of Corollary 31.2. For non-product measures that implication still uses Theorem 25.1.

Proof. On the short window the second-moment covariance bound is pointwise and integrates directly; the preprint-independent fallback instead splits on $\{\norm{A_t}_\op<2\}$ and pays the Brascamp–Lieb cap on the small complement. The calculation is carried out in Section Calculations.

The natural endpoint and the remaining universal-time gap¶

The fixed-time moment window between $c/\log^2n$ and $c/\log n$ is the consequence recorded in Theorem 31.3, with its explicit quadratic-chaos input. This does not silently strengthen the distinct published sup-over-time statement of Theorem 26.1. The scale $1/\log n$ is also the natural endpoint for covariance-only control, that is, the largest time scale such control can reach: the explicit product of centered one-sided exponentials has an eigenvalue that can reach order $\log n$ at times of order $1/\log n$ Klartag & Lehec, 2025, Remarks 62, 64 and Prop. 65. Thus the remaining gap for $\Xi_T^{(2)}$ is no longer an intermediate polylogarithmic interval; it is the passage from the sharp dimension-dependent early window to a universal time, precisely where cut-aware temporal alignment must replace covariance-only control.

Remark 31.3 (Heuristic — Expected failure of (V2) on universal windows for exponential products)

Let $\mu$ be the product of $n$ centered one-sided exponentials. Suppose, as the sharpness discussion of Klartag & Lehec, 2025, Remark 62 and Klartag, 2023 indicates, that with probability bounded below some coordinate variance reaches the scale of the Brascamp–Lieb cap, $A_t^{(i)}\asymp1/t$ , on a time window $[t_1,2t_1]$ with $t_1\asymp1/\log n$ . Then on that window $\E X_t^2\gtrsim t^{-2}$ , and

\int_{t_1}^{2t_1}\E X_t^2\dd t\ \gtrsim\ \frac1{t_1}\ \asymp\ \log n .

(31.8)

Hence on any universal window $[0,T_0]$ , the condition (V2) is expected to fail for exponential products, with $\int_0^{T_0}\E X_t^2\dd t\gtrsim\log n$ . We mark this as a heuristic: the cited statements establish inflation near the $1/\log n$ scale and describe the covariance estimate as “pretty much sharp”; the quantitative strength $\lmax\asymp1/t$ with non-negligible probability requires the sharpness example to be carried out in detail, which we have not done line by line.

Remark 31.4 (Consequence of the heuristic, stated conditionally)

If Remark 31.3 is correct, then the covariance-only argument of Corollary 31.2 cannot prove Assumption 28.1 for products on a universal window — even though both the assumption’s conclusion (KLS for products) and the per-cut estimates of Theorem 31.1 are true there. This is the precise product-model analogue of the insufficiency of the crude evaluation of $\Xi_T$ (Remark 33.2): cut-free covariance information saturates at a logarithm. The stress test thus shows, in refined form, that any proof of Assumption 28.1 for products must be cut-aware, and Theorem 31.1 exhibits the prototype mechanism — fixed per-coordinate budgets paid from the within-class variance $R_0$ — while Corollary 31.1 shows that the dual counterexample mechanism must likewise be cut-aware and high-complexity. The residue is a single question.

The residual: the adapted alignment problem¶

Combining Theorem 31.1 and Theorem 31.2, the anatomy of any counterexample to Assumption 28.1 within the product model is tightly constrained. To violate $\E\int_0^{T\wedge\tau}S_t\dd t\le MT$ for a given large $M$ and universal $T$ , a fixed balanced cut $E$ must satisfy: (a) $E$ depends on at least $MT$ coordinates (Theorem 31.1(i)); (b) at least of order $MT$ units of per-coordinate budget $\sum_i\E\int s\abs{Ge_i}^2$ — each coordinate contributing at most 1 over all time — must be spent inside the common short window $[0,T]$ ; (c) the expenditure must be concentrated, by Theorem 31.2, at times when $X_t$ is large, i.e.\ aligned with the rare inflation excursions of the independent coordinate processes; and (d) throughout, the mass must remain balanced ( $t<\tau$ ) and the source must not be matched by the damping $D_t$ , which the Riccati identity subtracts for free. The cut is fixed before the Brownian path; the inflating coordinate indices and the conditional means $a_{t,i}$ around which a two-tail configuration would have to center are random. The decisive question is whether (b)–(d) are simultaneously achievable.

Conjecture 31.1 (Adapted alignment problem for products — Not settled here · conj:product-alignment · Idea · Counterexample)

Let $\mu$ be a product of $n$ isotropic two-sided exponentials. There exist a universal $T_0$ and constants $C_0,C_1,\alpha<1$ such that for every $n$ and every balanced cut $E$ , writing $P_t^H$ for the coordinate projection onto $H_t=\{i:A_t^{(i)}\ge2\}$ and $P_t^L=I-P_t^H$ ,

\E\int_{I\cap[0,\tau]} S_t^H\dd t \ \le\ C_0\abs I+C_1\E\int_{I\cap[0,\tau]}r_t\dd t +\alpha\,\E\int_{I\cap[0,\tau]}D_t\dd t , \qquad I\subset[0,T_0] .

(31.9)

Here

S_t^H:=s_t\bigl(\norm{P_t^HG_tP_t^H}_\HS^2 +2\norm{P_t^LG_tP_t^H}_\HS^2\bigr) =S_t-s_t\norm{P_t^LG_tP_t^L}_\HS^2

(31.10)

counts every matrix entry incident to an inflated coordinate. This definition avoids an absorption loss: the simpler column mask counts each high–low entry only once, whereas the full source counts it twice. It also satisfies $S_t^H\le2s_t\sum_{i\in H_t}\abs{G_te_i}^2$ . The negation says that a fixed cut can spend $\Omega(1)$ -fractions of unboundedly many per-coordinate budgets inside the inflation excursions of those coordinates, within a common universal window, while staying balanced and underdamped. The displayed estimate, combined with Theorem 31.3 for the early $c/\log n$ window (conditional on the cited preprint) and a matching treatment of the non-inflated part at moderate times, would settle Remark 27.3 positively. Its failure — an explicit high-complexity cut achieving the alignment — would refute Assumption 28.1 on products, which satisfy KLS, since $S_t^H\le S_t$ ; it would bear on Assumption 28.2 only if the violating intervals could be taken to be prefixes on the tight window, and it would constrain no approach outside the fixed-cut one.

Remark 31.5 (Designated test family)

The first family one might try, the energy shell $\{\sum_i x_i^2\ge\theta\}$ , is too radial: its source is already covered on the early covariance window and does not interrogate the dangerous adapted alignment. A sharper family, and the natural place to look for a counterexample, is the balanced tail union

E_n=\{\max_i\abs{x_i}\ge a_n\} =\Bigl\{\sum_i\one_{\{\abs{x_i}\ge a_n\}}\ge1\Bigr\}, \qquad \mu(E_n)=\frac12.

(31.11)

It is permutation-symmetric but coordinate-selective, depends on all coordinates, and its complement is a product of one-dimensional truncated factors. Symmetry spreads the budget uniformly, while exchangeability lets a random inflating coordinate find the cut. A useful computation must measure the incident high-variance source $S_t^H$ in Conjecture 31.1, together with $r_t$ , $D_t$ , balance survival, and interval sweeps past $c/\log n$ . Such a computation would be evidence on a model in search of a counterexample, and would prove nothing about the all-cut estimate.

Methodological constraints from this chapter¶

Calculations¶

The long computations behind Theorem 31.1, Lemma 31.2 and Theorem 31.3. What each statement contributes, and the idea of each proof, are given where the statement is made; nothing is decided here that is not decided there. The proof of the bootstrap comparison is at the end of Chapter The fixed cut: the bootstrap.

Proof 31.5 (Proof of Theorem 31.1)

(i) Since $\mu_t$ is a product pathwise and $E$ is $J$ -measurable, Lemma 31.1 applies to $(\mu_t,E)$ at every time: $G_t$ is supported on the $J\times J$ block, so

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

(31.12)

The per-direction Carleson estimate Corollary 24.1 gives, for each fixed $i$ and every $T$ ,

\E\int_0^T s_t\abs{G_te_i}^2\dd t\le e_i^TR_0e_i=(R_0)_{ii}\le1,

(31.13)

using $R_0=A_0-B_0\preceq A_0=I$ . Sum over $i\in J$ and let $T\to\infty$ by monotone convergence.

(ii) The scalar Riccati identity Theorem 24.1 gives, after stopping and taking expectations within the regularity convention, $\E r_{t\wedge\tau}\le r_0+\E\int_0^{t\wedge\tau}S_s\dd s\le1+k$ , using $D\ge0$ and $r_0\le1$ (rank-one $B_0\preceq I$ ).

(iii) By (30.3), $\dd[p]_t=s_tr_t\dd t$ with $s_t\le\tfrac14$ , so

\E[p]_{T\wedge\tau}\le\frac14\int_0^T\E r_{t\wedge\tau}\dd t\le\frac{(1+k)T}4 .

(31.14)

Doob’s $L^2$ inequality for the exit of the nested coarse window gives $\Prob(\tau\le T)\le C_1(1+k)T$ with a universal $C_1$ ; choose $T_k=1/(2C_1(1+k))$ so that $\Prob(\tau>T_k)\ge\tfrac12$ , on which event $\min(p_{T_k},q_{T_k})\ge\tfrac13$ . The $T_k$ -uniform log-concavity of $\mu_{T_k}$ and the perimeter supermartingale then give, as in Theorem 30.1, $\mu^+(E)\ge\E\mu_{T_k}^+(E)\ge c\sqrt{T_k}\cdot\tfrac13\cdot\tfrac12\ge c'/\sqrt{1+k}$ .

Proof 31.6 (Proof of Lemma 31.2)

Expand $Y^TMY=\sum_iM_{ii}Y_i^2+\sum_{i\ne j}M_{ij}Y_iY_j$ . By independence and centering, all cross-covariances vanish: $\Cov(Y_i^2,Y_iY_j)=\E[Y_i^3]\E[Y_j]=0$ , $\Cov(Y_iY_j,Y_iY_k)=\E[Y_i^2]\E[Y_j]\E[Y_k]=0$ for $j\ne k$ , and disjoint pairs are independent. Hence, using $M_{ij}=M_{ji}$ so that each unordered pair $\{i,j\}$ contributes $\Var\bigl(2M_{ij}Y_iY_j\bigr)=4M_{ij}^2\sigma_i^2\sigma_j^2$ ,

\Var(Y^TMY)=\sum_iM_{ii}^2\Var(Y_i^2) +\sum_{i<j}4M_{ij}^2\sigma_i^2\sigma_j^2 =\sum_iM_{ii}^2\Var(Y_i^2)+2\sum_{i\ne j}M_{ij}^2\sigma_i^2\sigma_j^2 .

(31.15)

For one-dimensional log-concave $Y_i$ , the reverse Hölder inequalities (see e.g.\ Klartag & Lehec, 2025, Cor. 5) give $\E Y_i^4\le C_4\sigma_i^4$ with $C_4$ universal, so $\Var(Y_i^2)\le(C_4-1)\sigma_i^4$ . Both sums are dominated by $C_*\sum_{ij}M_{ij}^2\sigma_i^2\sigma_j^2=C_*\norm{A^{1/2}MA^{1/2}}_\HS^2$ with $C_*=\max(C_4-1,2)$ .

Proof 31.7 (Proof of Theorem 31.3)

On $t\le c_0/\log n$ , Corollary 26.1 with $p=2$ gives $\E\norm{A_t}_\op^2\le C_2$ , hence $\E X_t^2\le C_2$ . This pointwise estimate integrates over every subinterval and proves (V2). The consequences follow from Corollary 31.2.

For the preprint-independent fallback, on the event $\{\norm{A_t}_\op<2\}$ one has $X_t\le1$ . On the complement, of probability at most $e^{-1/(Ct)}$ by Theorem 26.1, the Brascamp–Lieb cap (23.10) gives $X_t\le t^{-1}$ , and therefore

\E X_t^2\le1+t^{-2}e^{-1/(Ct)}\le1+\sup_{u>0}u^2e^{-u/C}=1+(2C/e)^2=:K_0 .

(31.16)

References¶

Eldan, R. (2013). Thin Shell Implies Spectral Gap up to Polylog via a Stochastic Localization Scheme. Geometric and Functional Analysis, 23(2), 532–569. 10.1007/s00039-013-0214-y
Lee, Y. T., & Vempala, S. S. (2024). Eldan’s Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing. Annals of Mathematics, 199(3), 1043–1092. 10.4007/annals.2024.199.3.2
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). On Yuansi Chen’s Work on the KLS Conjecture. Lecture notes, Weizmann Institute of Science.