BK: reverse coefficient transfer - The KLS theorem and its methods

Part of the Balasubramanian–Kasiviswanathan proof, Chapter Balasubramanian–Kasiviswanathan: compatible integration; the reading order is on the full proofs page.

Overview. This reconstructs Proposition 7.1 of Balasubramanian & Kasiviswanathan, 2026, at commit 4837c33649ba2271f43c9684e9350ecbdd725f95. A terminal Appell polynomial is recovered from its lower-degree derivative. One matrix controls both the curvature normalization and the derivative norm. Ordered covariance moments and moving variance then transfer the estimate back to the original measure.

Dependencies. We use Definition 11.1, Definition 11.2, Proposition 11.1, Lemma 11.1, Lemma 11.4, and Lemma 11.5. No KLS, BKL, or SZ v2 assertion is used. The integration product bound and lower-degree constants below are explicit hypotheses, not conclusions imported from any of those proofs.

Theorem 155.1 (Quantified reverse transfer)

Fix integers $d\ge2$ and $1\le q\le d-1$ , and $0<\eta\le1$ . Set $\delta=\eta^2/d^2$ . Suppose $M_q\ge0$ is finite and, for every centered regular log-concave law $\nu$ in every dimension with covariance at most $I$ and curvature at least $\delta I$ ,

\|J_0^\nu J_1^\nu\cdots J_{q-1}^\nu\|_{\rm op}\le M_q.

(155.1)

Suppose finite $C_1,\ldots,C_{d-1}\ge0$ satisfy $c_k(\nu)\le C_k$ for every isotropic log-concave law in every dimension and $1\le k<d$ . Put $\Sigma_d=\sum_{k=2}^{d-1}(d-k+1)C_kC_{d-k+1}$ , with empty sum zero. Then

c_d^*\le e^{3\eta}\left[(1+\eta)^{q/2}M_qC_{d-q} +\frac{\eta}{d^2}\Sigma_d\right].

(155.2)

This is Proposition 11.2, with $c_d^*$ the supremum over all centered log-concave laws of covariance at most identity in all dimensions.

Proof 155.1

For a regular law $\nu$ , let $p$ be a degree- $d$ Appell polynomial. All derivatives $D^jp$ , $0\le j\le q$ , are centered, since $q<d$ . They are compatible symmetric fields, and polynomial integrability places them in the required weighted Sobolev domains. The centered-primitive identity of Proposition 11.1 applied successively gives

p=J_0^\nu J_1^\nu\cdots J_{q-1}^\nu D^qp, \qquad E_\nu p^2\le M_q^2 E_\nu\|D^qp\|^2.

(155.3)

Here the final derivative is centered too; this is precisely where $q<d$ is required. There is no assertion of this product identity on constant fields.

The same inequality holds for a full-dimensional centered, possibly nonsmooth law with covariance at most $I$ and curvature at least $\delta I$ . Indeed Lemma 11.1 gives regular laws with these exact same two bounds and convergence of all fixed moments. Apply the inequality to their Appell polynomials with fixed leading tensor. Coefficients of an Appell polynomial are polynomials in moments through degree $d$ , obtained by formal reciprocal expansion with constant coefficient one. Its squared norm and that of its derivative therefore depend continuously on moments through $2d$ . Passing to the limit proves the assertion without approximating any integration operator. The applicability condition $\delta\le1$ holds here.

Let now $\mu$ have mean $m$ , positive covariance $A$ , and positive curvature matrix $B$ . Set

M=A+\delta B^{-1},\qquad Y=M^{-1/2}(X-m),\qquad \nu=\mathcal L(Y).

(155.4)

Then $\operatorname{Cov}(\nu)=M^{-1/2}AM^{-1/2}\preceq I$ . Moreover $M\succeq\delta B^{-1}$ gives $B\succeq\delta M^{-1}$ , so the curvature of $\nu$ is at least $M^{1/2}BM^{1/2}\succeq\delta I$ . For $\bar p(y)=p(m+M^{1/2}y)$ the Appell affine identity preserves all centering conditions. The derivative chain rule gives $D_y^q\bar p=(M^{1/2})^{\otimes q}D_x^qp$ . Consequently

E_\mu p^2\le M_q^2 E_\mu\langle D^qp,M^{\otimes q}D^qp\rangle.

(155.5)

This argument also applies to nonsmooth full-dimensional $\mu$ by the preceding approximation after the coordinate change.

Start with a compactly supported isotropic law and run the localization of Lemma 11.4. Fix $T\in\operatorname{Sym}^d\mathbb R^n$ , and set $p_t=P_d^{\mu_t}[T]$ . For every positive semidefinite matrix $M$ ,

E_t\langle D^qp_t,M^{\otimes q}D^qp_t\rangle \le(d!)^2C_{d-q}^2 \langle T,(M^{\otimes q}\otimes A_t^{\otimes(d-q)})T\rangle.

(155.6)

To see every tensor factor and factorial, put $S=((M^{1/2})^{\otimes q}\otimes(A_t^{1/2})^{\otimes(d-q)})T$ . For each ordered derivative index tuple $I$ , the remaining slice $S_I$ is symmetric of rank $d-q$ . Differentiation gives $d!/(d-q)!$ , and the coefficient bound in isotropic coordinates gives $(d-q)!C_{d-q}$ . Squaring and summing these scalar bounds over $I$ gives the displayed formula, since $\sum_I\|S_I\|^2=\|S\|^2$ . No symmetry between the two groups of slots is required, and no commutation of $M$ with $A_t$ is asserted.

For $t>0$ the curvature is at least $\Lambda_t\succ0$ . Choose $M=A_t+\delta\Lambda_t^{-1}$ and apply both preceding inequalities. With $V(t)=\mathbb E E_tp_t^2$ , the ordered mixed covariance estimate gives

V(t)\le(d!)^2C_{d-q}^2M_q^2(1+\delta/t)^q e^{4d^2t}\|T\|^2.

(155.7)

Substitute its square root in Lemma 11.5. For $T\ne0$ , divide by $d!\|T\|$ to obtain

\frac{\sqrt{V(0)}}{d!\|T\|} \le e^{(2d^2+d+1)t} \left[(1+\delta/t)^{q/2}M_qC_{d-q}+t\Sigma_d\right].

(155.8)

Set $t=\eta/d^2$ . Then $\delta/t=\eta$ and $(2d^2+d+1)t\le3\eta$ for $d\ge2$ . Supremizing in $T$ proves the claimed bound for compactly supported isotropic laws.

For an arbitrary isotropic log-concave $X$ , condition on $|X|\le R$ . For large $R$ , its conditional law has positive definite covariance $A_R$ ; its mean $m_R$ tends to zero and $A_R\to I$ . All polynomial moments converge by dominated convergence, using finite log-concave moments. The whitened vector $A_R^{-1/2}(X_R-m_R)$ is compactly supported, isotropic and log-concave, and all its fixed moments converge to those of $X$ . The degree- $d$ Appell Gram matrix in a fixed orthonormal basis of $\operatorname{Sym}^d\mathbb R^n$ consequently converges entrywise, hence in operator norm. Its largest eigenvalue is $(d!c_d)^2$ , so $c_d$ converges. This passes the same bound to every isotropic law.

Finally, if a centered law has covariance at most $I$ , restrict to its linear support. Unless it is a point mass, it is the image $BY$ of an isotropic law on that support with $\|B\|_{\rm op}\le1$ . The Appell transformation law sends $T$ to $(B^T)^{\otimes d}T$ , whose norm is at most $\|T\|$ . Thus the same bound holds. For a point mass all positive-degree Appell variances vanish. Taking the supremum over all dimensions and all such laws finishes the proof.

Fences respected. No bounded_by edge is assigned to this new proposition. The brief’s uniform-admissibility warning is met by requiring one $M_q$ for all laws at the prescribed $\delta$ and one $C_k$ for each lower degree over all isotropic laws. The statement is an implication and does not silently provide these bounds. It makes no comparison of existing fixed-cut or gate-zero routes and does not use a conclusion of KLS to initialize the coefficients.

Dependencies and scope. The argument uses preservation of both exact bounds by the regular approximation input, the centered domain at the last integration, the two separate tensor metrics, the source’s $\eta/d^2$ error term, and the truncation and affine-support passages. The approximation and integration inputs are proved separately; this dossier does not replace their proofs or independent reviews.

References¶

Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728v1