Consequences of Letwin's quadratic estimate - The KLS theorem and its methods

Part of the shared technical foundations, Chapters The static quadratic-chaos input and the two-tail obstruction and The fixed cut: product stress test; the reading order is on the full proofs page.

Overview. Whitening and duality convert the quadratic-chaos theorem into a bound on the two-color contrast. Unwhitening costs exactly the square of the largest covariance eigenvalue; removing the off-balance correction gives the constant in the manuscript. Applied separately to every posterior, this gives the pathwise covariance reduction and its integrated version. Product posteriors admit the same argument using the certified product quadratic-chaos estimate, without Letwin’s input.

Logical scope. The unconditional statement Corollary 25.1 uses Theorem 25.1 as a proof dependency. Its certification therefore requires that input to be established. The general-measure branches of Theorem 31.2 and Corollary 31.2 retain the explicit antecedent Theorem 25.1. Their product branches use no such antecedent. This dossier does not certify Letwin’s source theorem.

Theorem 106.1 (Whitened two-color control)

Let $\nu$ be centered log-concave with positive-definite covariance $A$ , and let $E$ have mass $p\in(0,1)$ . Put $q=1-p$ , $s=pq$ , $\delta=m^E-m^{E^c}$ , $G=\Sigma^E-\Sigma^{E^c}$ and $K=G+(q-p)\delta\delta^T$ . Then the conclusion of Corollary 25.1 is

s\|A^{-1/2}KA^{-1/2}\|_{\mathrm{HS}}^2\le8, \qquad s\|K\|_{\mathrm{HS}}^2\le8\|A\|_{\mathrm{op}}^2.

(106.1)

If $p\in[1/3,2/3]$ , then $s\|G\|_{\mathrm{HS}}^2\le17\|A\|_{\mathrm{op}}^2$ ; at $p=1/2$ the constant is 8.

Proof 106.1

Write $Y=A^{-1/2}X$ for $X\sim\nu$ , and transport the cut by this invertible map. Its mass does not change. Conditional means and covariances transform linearly and by congruence respectively, so the new contrast is $H=A^{-1/2}KA^{-1/2}$ . All moments used below exist because log-concave probabilities have finite polynomial moments. Let $g=(\mathbf1_E-p)/\sqrt{s}$ on the whitened probability space. Then $\mathbb Eg=0$ , $\mathbb Eg^2=1$ . For symmetric $N$ , the certified identity Proposition 32.1 gives

\sqrt{s}\langle H,N\rangle_{\mathrm{HS}} =\mathbb E\big[g(Y^TNY-\operatorname{Tr}N)\big].

(106.2)

Cauchy--Schwarz and Theorem 25.1 give $s\langle H,N\rangle^2\le8\|N\|_{\mathrm{HS}}^2$ . If $H\ne0$ , take $N=H/\|H\|_{\mathrm{HS}}$ ; for $H=0$ the conclusion is immediate. This proves the whitened estimate. For every matrix $L$ , $\|A^{1/2}LA^{1/2}\|_{\mathrm{HS}} \le\|A^{1/2}\|_{\mathrm{op}}^2\|L\|_{\mathrm{HS}}$ , as follows by applying the operator norm bound successively to left and right multiplication. With $L=H$ this proves the second estimate.

Covariance decomposition gives $A=p\Sigma^E+q\Sigma^{E^c}+s\delta\delta^T$ , so $B=s\delta\delta^T\preceq A$ . Since $B$ has rank at most one, $r=s|\delta|^2\le\|A\|_{\mathrm{op}}$ . Consequently

\begin{aligned} s\|G\|_{\mathrm{HS}}^2 &\le2s\|K\|_{\mathrm{HS}}^2+2s(q-p)^2|\delta|^4\\ &\le16\|A\|_{\mathrm{op}}^2+\frac{2(q-p)^2}{s}r^2. \end{aligned}

(106.3)

On the coarse window, $s\ge2/9$ and $|q-p|\le1/3$ , hence the last coefficient is at most one. This gives 17. At exact balance $K=G$ , so the previous estimate gives 8 without this loss.

Proof 106.2

For any finite posterior realization, the likelihood $\exp(c_t\cdot x-t|x|^2/2)$ is strictly positive on the support of the initial law. An isotropic initial law is not supported on a proper affine hyperplane, and equivalence under positive likelihood preserves this property. Thus its posterior covariance is positive definite at finite times. At $t>0$ the Gaussian factor makes every polynomial moment finite; at zero this follows from log-concavity. No singular inverse or infinite-time posterior is being used. The posterior, centered at its mean, is log-concave, and translating both law and cut does not change $G$ or $K$ .

Apply the preceding two-color bound to that posterior: on the coarse window $S_t\le17\|A_t\|_{\mathrm{op}}^2$ . Since $\|A_t\|_{\mathrm{op}}\le1+X_t$ and $(1+X_t)^2\le2(1+X_t^2)$ , one can take $C=34$ in the general branch. This is a deterministic inequality for each admissible posterior, hence holds pathwise without taking a supremum of exceptional events over times.

For the independent product branch, the posterior density factorizes:

\mu_t(dx)=\bigotimes_i \frac{e^{c_{t,i}x_i-tx_i^2/2}\mu_i(dx_i)} {\int e^{c_{t,i}z-tz^2/2}\mu_i(dz)}.

(106.5)

Center its coordinates. By Lemma 31.2, for every symmetric $M$ , the centered posterior $Y$ with diagonal covariance $A_t$ satisfies $\operatorname{Var}(Y^TMY)\le C_*\|A_t^{1/2}MA_t^{1/2}\|_{\mathrm{HS}}^2 \le C_*\|A_t\|_{\mathrm{op}}^2\|M\|_{\mathrm{HS}}^2$ . The same color identity and duality just used therefore give $s_t\|K_t\|_{\mathrm{HS}}^2\le C_*\|A_t\|_{\mathrm{op}}^2$ . The off-balance calculation gives $S_t\le(2C_*+1)\|A_t\|_{\mathrm{op}}^2 \le2(2C_*+1)(1+X_t^2)$ , as required. Product structure of the cut itself is not assumed.

Proof 106.3

Before its coarse exit time $\tau$ , the cut is in the window where the preceding pathwise estimate applies. All integrands are nonnegative, so Tonelli and the premise give

\mathbb E\int_{I\cap[0,\tau]}S_t\,dt \le C\int_I(1+\mathbb EX_t^2)\,dt \le C(1+K)|I|.

(106.7)

This is exactly the claimed all-cut bound, without any damping or $r$ term. It holds for each fixed cut, with a constant independent of the cut. For products use the unconditional branch instead. Furthermore each coordinate posterior is driven by its own one-dimensional localization: the drift of $c_{t,i}$ is its posterior mean, a function only of $(t,c_{t,i})$ , and the Brownian coordinates are independent. Thus the variance processes are independent one-dimensional processes, and $\|A_t\|_{\mathrm{op}}$ is their maximum. The condition contains no cut variable.

Hypotheses and fences. Positive definiteness is used only for static whitening and follows from isotropy and positive likelihood on finite posterior paths. Balance is used only for the bounded off-balance correction. Product independence is used only in the preprint-independent branch. None of these targets has a bounded_by edge. The covariance loss is retained: no intrinsic estimate is mistaken for a universal Euclidean source bound, and a dimension-dependent time horizon is not a universal Carleson horizon.