BKL: the all-order cumulant bound - The KLS theorem and its methods

Part of the Bizeul–Klartag–Lehec proof, Chapter Bizeul–Klartag–Lehec: cumulants and suspension; the reading order is on the full proofs page.

Overview. This dossier reconstructs Theorem 4.1 and Section 6 of Bizeul et al., 2026, version 1. The induction proves a static bound and an integrated next-order bound together. In a split cumulant contraction the factor containing the distinguished vector uses the integrated estimate; the other factor uses the static estimate. The resulting factorial cancellation closes the induction without a dimension factor.

Author. plan_framework researcher, unknown, 2026-10-06.

Dependencies. Definition 8.1, Lemma 8.3 and Lemma 8.4. The third-order input used in the dynamics is Proposition 26.1 with its antecedent discharged by Theorem 25.1. No KLS estimate, suspension argument or tilt-average criterion is used.

Proof 134.1

Fix $n$ . All constants chosen below are independent of this choice. Initially restrict to compactly supported isotropic log-concave laws. Let $C$ be the universal constant in Lemma 8.4, and choose

K\ge\max\{5,9C,64\},\qquad b_r=K^{r-1}((r-1)!)^2\quad(r\ge2).

(134.2)

We prove, simultaneously for every such law and every unit vector $u$ , that

\mathcal E_r(0)+\tfrac12\int_0^\infty\mathbb E\mathcal E_{r+1}(t)\,dt \le b_r\qquad(r\ge2). \tag{A}

(134.3)

For $r=2$ , $\mathcal E_2(0)=1$ and $\mathbb E\mathcal E_3(t)\le8e^{-t}$ by Lemma 8.3, so the left side is at most $5\le b_2$ . This initializes the integrated estimate needed when $r=3$ .

Suppose (A) is proved at all orders $2\le r<m$ , where $m\ge3$ . It supplies two different consequences:

\int_0^\infty\mathbb E\mathcal E_{r+1}(t)\,dt\le2b_r, \qquad R_r^\nu\le b_rI \quad\hbox{for every compactly supported isotropic log-concave }\nu. \tag{B}

(134.4)

The first concerns the process started at the law presently under study. The second can be applied to its whitened posterior at every time. Thus

|\kappa_r(t)(v)|_t^2\le b_r\langle A_tv,v\rangle \quad(2\le r<m),\qquad I_m:=\int_0^\infty\mathbb E\mathcal E_m(t)\,dt\le2b_{m-1}. \tag{C}

(134.5)

It remains to bound the split tensor $L_m$ . For a fixed cardinality $a\in\{2,\ldots,m-2\}$ , one summand, after permuting the nondistinguished slots, is

P_a(h_2,\ldots,h_m)= \left\langle\kappa_{a+1}(t)(u,h_2,\ldots,h_a,\cdot), A_t^{-1}\kappa_{m-a+1}(t)(h_{a+1},\ldots,h_m,\cdot)\right\rangle.

(134.6)

To compute its weighted norm put $w_j=A_t^{-1/2}e_j$ and first fix the indices $i_2,\ldots,i_a$ . Let $z=\kappa_{a+1}(t)(u,w_{i_2},\ldots,w_{i_a},\cdot)$ . Summing over the remaining $m-a$ indices and applying (C) gives

\begin{split} &\sum_{i_{a+1},\ldots,i_m} P_a(w_{i_2},\ldots,w_{i_m})^2\\ &\qquad=|\kappa_{m-a+1}(t)(A_t^{-1}z)|_t^2 \le b_{m-a+1}\langle z,A_t^{-1}z\rangle =b_{m-a+1}\sum_k\kappa_{a+1}(t)(u,w_{i_2},\ldots,w_{i_a},w_k)^2. \end{split}

(134.7)

The order $m-a+1$ is at most $m-1$ , so the use of (C) is legitimate. Now sum over $i_2,\ldots,i_a$ , and integrate in time and probability. By (B) at order $a$ , this gives

|P_a(t)|_t^2\le b_{m-a+1}\mathcal E_{a+1}(t),\qquad \int_0^\infty\mathbb E|P_a(t)|_t^2\,dt\le2b_ab_{m-a+1}. \tag{D}

(134.8)

For each $a$ there are $\binom{m-1}{a-1}$ subsets containing the distinguished index. Each yields a permutation of the other tensor indices and has the same weighted norm. Apply the triangle inequality in the Hilbert space of whitened tensors over time and probability. It follows that

\begin{split} \sqrt{J_m}&:= \left(\int_0^\infty\mathbb E|L_m(t)(u)|_t^2\,dt\right)^{1/2}\\ &\le\sqrt2\sum_{a=2}^{m-2}\binom{m-1}{a-1}\sqrt{b_ab_{m-a+1}}. \end{split}

(134.9)

There is no dimension factor in this estimate. Substitution of the definition of $b_r$ shows, term by term, that

\binom{m-1}{a-1}\sqrt{b_ab_{m-a+1}} =\frac{(m-1)!}{(a-1)!(m-a)!} K^{(m-1)/2}(a-1)!(m-a)!=\sqrt{b_m}.

(134.10)

There are $m-3$ terms; for $m=3$ the sum is empty. Thus $\sqrt{J_m}\le\sqrt2(m-3)\sqrt{b_m}$ . Both integrability hypotheses of Lemma 8.4 have now been proved, with no use of (A) at order $m$ . That lemma and (C) imply

\begin{split} \mathcal E_m(0)+\tfrac12\int_0^\infty\mathbb E\mathcal E_{m+1}(t)\,dt &\le2Cm^2b_{m-1}+4(m-3)\sqrt{b_{m-1}b_m}\\ &=b_m\left(\frac{2C}{K}\frac{m^2}{(m-1)^2} +\frac4{\sqrt K}\frac{m-3}{m-1}\right)\le b_m. \end{split}

(134.11)

Indeed $m^2/(m-1)^2\le9/4$ , so the first term in parentheses is at most $1/2$ when $K\ge9C$ , and the second is at most $1/2$ when $K\ge64$ . This closes the simultaneous induction. Since $\mathcal E_m(0)=|\kappa_m^\mu(u)|^2$ , its static part proves the theorem for compactly supported laws.

Removal of compact support. Let now $X$ have any isotropic log-concave law $\mu$ . For sufficiently large $R$ let $X_R$ have its conditional distribution on the convex ball $\{|x|\le R\}$ . These laws are log-concave and full dimensional. Their means $a_R$ converge to zero and covariance matrices $A_R$ converge to $I$ . To justify convergence of all higher moments, each coordinate of $X$ is a variance-one log-concave marginal, and the elementary tail bound proved in Lemma 8.3 shows that every coordinate has every absolute moment. Thus $|X|$ has every fixed moment as well. Dominated convergence, followed by division by $\mathbb P(|X|\le R)\to1$ , gives convergence of all mixed moments of $X_R$ .

Set $Y_R=A_R^{-1/2}(X_R-a_R)$ . These are compactly supported isotropic log-concave vectors, and $A_R^{-1/2}\to I$ . Expanding each fixed mixed moment of $Y_R$ as a finite polynomial in the coefficients of $A_R^{-1/2}$ , $a_R$ and mixed moments of $X_R$ proves its convergence to that of $X$ . Cumulant entries are finite polynomials in these moments by the moment-cumulant formula. Therefore $\kappa_m^{Y_R}$ converges entrywise to $\kappa_m^\mu$ at every fixed order $m$ . In fixed dimension its Hilbert–Schmidt norm and every directional contraction converge. The compact-support estimate passes to the limit with the same universal $K$ . Homogeneity removes the restriction $|u|=1$ , including $u=0$ . No stochastic process for a noncompact initial law is needed in this limiting argument.

Fences respected. There is no assigned bounded_by fence. Constants in the final induction do not depend on support radius, dimension, cumulant order or the regularity of the density. Dimension-dependent moment constants are used only to justify finite-time expectations and moment convergence, never in the recurrence for $b_m$ .

References¶

Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a Proof of the Kannan–Lovasz–Simonovits Conjecture. https://arxiv.org/abs/2610.05474v1