BK: the explicit degree induction - 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. A convolution estimate controls the lowering drift. For bounded initial degrees the quadratic integration seed and a small localization time close the induction. In larger degrees, finitely many lower coefficients feed the uniform power lemma at a rate strictly smaller than the target radius; this gain absorbs the coordinate change and weight ratio. Every estimate below is analytic. This reconstructs Theorem 8.1 and Appendix E of Balasubramanian & Kasiviswanathan, 2026, pinned at commit 4837c33649ba2271f43c9684e9350ecbdd725f95.

Dependencies and explicit inputs. We use the coefficients and operators of Definition 11.1 and Definition 11.2. The graded calculus Proposition 11.1 is an explicit input: for every centered regular law of covariance at most $I$ and curvature at least $aI$ , there are bounded operators $J,L$ with

J^*J\preceq g_a(JJ^*+LL^*),\quad g_a(u)=u/(1+au),\qquad \|J^{j-1}L\|\le c_j(\nu),

(156.1)

and $J_0\cdots J_{q-1}$ is a block of $J^q$ . We use Lemma 11.3 and Corollary 11.2. The reverse-transfer input Proposition 11.2 is used in the following exact form: for integers $d\ge2$ , $1\le q<d$ , $0<\eta\le1$ , $\delta=\eta^2/d^2$ , and finite constants $C_k\ge c_k^*$ for $1\le k<d$ , if

\|J_0^\nu\cdots J_{q-1}^\nu\|\le M_q<\infty

(156.2)

for every centered regular law with covariance at most $I$ and curvature at least $\delta I$ , then

c_d^*\le e^{3\eta}\left((1+\eta)^{q/2}M_qC_{d-q} +\frac{\eta}{d^2}\Sigma_d\right),\qquad \Sigma_d=\sum_{k=2}^{d-1}(d-k+1)C_kC_{d-k+1}.\tag{1}

(156.3)

In particular this input includes passage to all log-concave laws in $c_d^*$ ; no limiting regularity claim is silently added here. No BKL, SZ v2, KLS conclusion, or already uniform exponential Appell bound is used.

Proof 156.1

Put $w(j)=(j+1)^{-4}$ for $j\ge0$ , $C_w=4$ , and $D_0=10^7$ . We first verify the convolution estimate

\sum_{k=1}^s\frac{w(k)w(s-k)}{w(s)}\le C_w\qquad(s\ge1).\tag{2}

(156.5)

The endpoint $k=s$ contributes one. For $1\le k\le s/2$ , $(s+1)/(s-k+1)\le2$ , so that term is at most $16/(k+1)^4$ . Pairing the interior terms about the midpoint (allowing double counting there) bounds the left side by

1+32\sum_{j=2}^\infty j^{-4} \le1+32\left(\frac1{16}+\frac1{81}+\int_3^\infty x^{-4}\,dx\right) =\frac{307}{81}<4.

(156.6)

This also covers $s=1$ , when there are no interior terms.

The first coefficient is the standard deviation of a linear functional, so $c_1^*\le1<R/16=\beta_1$ . Fix $d\ge2$ and suppose inductively $c_k^*\le\beta_k$ for every $1\le k<d$ . In (1) set $C_k=\beta_k$ . Every degree appearing in $\Sigma_d$ is less than $d$ . Since $d-k+1\le d$ , (2) at $s=d+1$ gives

\Sigma_d\le dC_wR^{d+1}w(d+1).\tag{3}

(156.7)

For $d=2$ the sum is empty and this bound remains true.

Initial degrees. Suppose $2\le d\le D_0$ . Set

q=1,\qquad\eta=\frac{d}{100R},\qquad \delta=\frac{\eta^2}{d^2}=(100R)^{-2}.

(156.8)

Then $0<\eta\le1/1000<1/100$ . For every law in the curvature class required by (1), the quadratic seed supplies

\|J_0\|\le\|J\|\le M_1:=(2/\delta)^{1/6} =[2(100R)^2]^{1/6}\le10\sqrt R.

(156.9)

The last inequality follows by sixth powers: $20000R^2\le10^6R^3$ . Also

\frac{\beta_{d-1}}{\beta_d}= \frac1R\left(\frac{d+1}{d}\right)^4<\frac{16}{R}.

(156.10)

By (3) and monotonicity of $w$ ,

\frac{\eta\Sigma_d}{d^2\beta_d} \le\frac{\eta C_wR}{d}=\frac1{25}.

(156.11)

Divide (1) by $\beta_d$ . The elementary inequalities $e^x\le(1-x)^{-1}$ for $0\le x<1$ and $\sqrt{1+\eta}\le1+\eta$ imply $e^{3\eta}<21/20$ and $\sqrt{1+\eta}<21/20$ . As $160/\sqrt R=2/125$ , it follows that

\frac{c_d^*}{\beta_d} \le e^{3\eta}\left(\sqrt{1+\eta}\frac{160}{\sqrt R}+\frac1{25}\right) <\frac{21}{20}\left(\frac{21}{20}\frac2{125}+\frac1{25}\right) =\frac{1491}{25000}<1.

(156.12)

This closes every initial degree, using only already bounded lower degrees.

Degrees beyond the cutoff. Suppose $d>D_0$ . Choose

q=\lfloor d/2\rfloor,\quad D=\lfloor\sqrt d\rfloor,\quad \eta=\frac{d^{-1/2}}{100C_wR},\quad \delta=\frac{d^{-3}}{(100C_wR)^2},\quad B=R^2e^{-1/(D+1)}.

(156.13)

These satisfy $1\le q<d$ , $2\le D<d$ , $0<\eta<1/100$ and $\delta=\eta^2/d^2$ . For every regular law in the curvature class of (1), the calculus input and induction give the observation bounds

\|J^{j-1}L\|\le c_j(\nu)\le c_j^*\le \gamma_j:=\frac{R^j}{(j+1)^4}\qquad(2\le j\le D).

(156.14)

The strict hypothesis of the power lemma is satisfied, because $$\delta B^{D+1} =\frac{R^{2D}}{e(100C_w)^2d^3}

\frac{R^{2D}}{(D+1)^8}=\gamma_D^2.\tag{4}$ $Indeed$ D+1>\sqrt d $implies$ (D+1)^8>d^4 $, while$ e(100C_w)^2<3\cdot400^2=480000<D_0<d $. The elementary bound$ e<3 $follows, for example, from the factorial series and$ k!\ge2^{k-1} $for$ k\ge1 $, with a strict inequality for$ k\ge3$.

The same power lemma, keeping its common prefactor, gives

\begin{aligned} Q_D(B)&=1+\sum_{j=1}^{D-2}\frac{je^{(j+1)/(D+1)}}{(j+2)^8}\\ &\le1+e\sum_{j=1}^\infty(j+2)^{-7} \le1+\frac e{384}<2. \end{aligned}

(156.15)

Consequently one uniform bound valid for the entire curvature class is

\|J_0\cdots J_{q-1}\|\le\|J^q\| \le M_q:=\sqrt2R^q\exp\left(-\frac q{2(D+1)}\right).\tag{5}

(156.16)

Thus (1) applies without any dependence of $M_q$ on dimension or law.

The principal term in (1), divided by $\beta_d$ , equals

\sqrt2\frac{w(d-q)}{w(d)} \exp\left(3\eta+\frac q2\log(1+\eta)-\frac q{2(D+1)}\right).

(156.17)

Since $d-q\ge d/2$ , the weight ratio is at most 16. Using $\log(1+\eta)\le\eta$ and $\sqrt2<2$ , the term is at most

32\exp\left(3\eta+q\left[\frac\eta2-\frac1{2(D+1)}\right]\right).

(156.18)

Now $D+1\le2\sqrt d$ and hence $\eta(D+1)\le2/(100C_wR)<1/2$ ; the bracket is at most $-1/[4(D+1)]$ . Since $q\ge d/3$ ,

\text{principal ratio}\le 32\exp\left(\frac3{100}-\frac{\sqrt d}{24}\right)<\frac14.

(156.19)

For the last strict estimate, $d>D_0>10^6$ gives $\sqrt d>1000$ ; the exponent is less than -7, and the series for $e$ gives $e>2$ , so $32e^{-7}<1/4$ .

Finally (3) bounds the drift ratio, including its exponential factor, by

\frac{e^{3\eta}\eta\Sigma_d}{d^2\beta_d} \le\frac{e^{3\eta}\eta C_wR}{d} =\frac{e^{3\eta}}{100d^{3/2}}<\frac1{50}.

(156.20)

Here $e^{3\eta}<(1-3/100)^{-1}<2$ and $d\ge1$ . The two ratios sum to less than one, proving $c_d^*<\beta_d$ . The induction holds for every integer $d$ , with the single radius $R$ fixed before choosing dimension, law, or degree.

Fences respected. No separate bounded_by node is proposed. The quantifiers enforce one radius for all degrees, dimensions and laws; the argument never starts from merely degreewise finite constants. Curvature is positive at each invocation of the power lemma, its strict last-observation condition is checked in (4), and only degrees below $d$ feed the induction. Degenerate-support laws are included solely through the explicit scope of the reverse-transfer input and the elementary base case. The proof does not use a KLS-equivalent conclusion as its initialization.

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