The stopped source bound - The KLS theorem and its methods

Part of the fixed-eigenfunction mechanism, Chapter The fixed eigenfunction: following one eigenfunction through localization; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 21.3 as Theorem D17.1, using the certified quadratic-chaos theorem Theorem 25.1. For a regular approximant and a unit-variance fixed test, the source is integrated only up to the covariance exit time $\tau_L$ , and the bound is $\E\int_0^{T\wedge\tau_L}\norm{H_t}_{\HS}^2\dd t\le8L^2T$ ((D17.7)). The proof has three parts: a pathwise whitened duality bound from the Letwin inequality, unwhitening only where $\norm{A_t}_\op<L$ , and integration against the posterior variance budget. No moment of $\norm{A_t}_\op$ is used.

Posterior facts: the tilt (D17.3) is the conditional law ((D17.9)). It is strongly log-concave, with positive-definite covariance and all moments.
$H_t$ is defined by an absolutely convergent integral off a $\dd t\otimes\dd\Prob$ -null set.
Whitening the posterior produces an isotropic log-concave vector. Duality over symmetric $M$ together with Theorem 25.1 gives the pathwise bound (D17.8).
Before $\tau_L$ , the ideal property of the Hilbert–Schmidt norm unwhitens step 3, giving (D17.16).
The variance budget $\E v_t\le1$ ((D17.17)) and Tonelli integrate step 4 to (D17.7).

Refined statement. This dossier proves the stopped, unweighted source bound using Theorem 25.1 as a proved dependency. The statement is uniform in the dimension and in the regularization: the constant $8L^2$ contains no $n$ , no $\varepsilon$ , and no property of the test beyond its variance. No unstopped moment of $\norm{A_t}_\op$ is used, and no independence between the posterior variance and the covariance operator norm is asserted anywhere.

Setting. Throughout, $\mu$ is a regular approximant: a probability measure

\dd\mu(x)=Z^{-1}e^{-V(x)}\dd x \qquad\text{on }\R^n, \qquad V\in C^\infty(\R^n),\qquad \nabla^2V\succeq\varepsilon I_n,\ \varepsilon>0,

(D17.1)

as in the regular class of Section Setup and the exact fixed-function SDE. (Centering and isotropy are part of that class but are not used in this lemma; the proof only uses smoothness and strong log-concavity. We record this so the lemma can be consumed verbatim after the restart of the companion dossier, where the posterior prior is no longer isotropic.) Take $X\sim\mu$ , an independent standard Brownian motion $B^{\mathrm{obs}}$ , and the planted observation channel

c_t=tX+B_t^{\mathrm{obs}}, \qquad \mathcal F_t=\sigma(c_s:0\le s\le t).

(D17.2)

Define the pathwise posterior kernel by the explicit tilt formula

\dd\mu_t(x) =\frac{\exp\bigl(c_t\cdot x-t\abs{x}^2/2\bigr)} {\int\exp\bigl(c_t\cdot y-t\abs{y}^2/2\bigr)\dd\mu(y)}\,\dd\mu(x), \qquad t\ge0,

(D17.3)

and write $\E_t$ for integration against $\mu_t$ ,

m_t=\E_tf,\qquad a_t=\E_tX,\qquad v_t=\Var_{\mu_t}(f),\qquad A_t=\Cov_{\mu_t}(X),

(D17.4)

g_t=\E_t[(f-m_t)(X-a_t)], \qquad H_t=\E_t\bigl[(f-m_t)(X-a_t)^{\otimes2}\bigr],

(D17.5)

for a fixed test $f\in L^2(\mu)$ . When the defining integral of $H_t$ fails to converge absolutely we set $H_t=0$ ; Step 1 of the proof shows this happens only on a $\dd\Prob\otimes\dd t$ -null set, so the convention does not affect any integral below. For $L\ge1$ put

\tau_L=\inf\bigl\{t\ge0:\norm{A_t}_\op\ge L\bigr\}, \qquad \inf\emptyset=\infty .

(D17.6)

Only the elementary implication $t<\tau_L\Rightarrow\norm{A_t}_\op<L$ , immediate from the definition of the infimum, is used; no stopping-time property of $\tau_L$ is needed in this dossier.

Quadratic input and applicability. The certified Theorem 25.1 states $\Var(Y^TMY)\le8\norm M_{\HS}^2$ for every isotropic log-concave law on $\R^n$ , in every dimension, and every symmetric matrix $M$ . Its all-law conclusion is not restricted to the bounded-support regular class used in the moment-map proof. The centered and whitened posterior in Step 2 is an isotropic log-concave law, so this is exactly the needed input with the same constant 8.

Proof D17.1

Step 0: posterior facts. Conditional on $X=x$ , the path $(c_s)_{s\le t}$ is a Brownian motion with constant drift $x$ ; by the Cameron–Martin theorem its law on path space has, with respect to the Wiener measure, the density $\exp\bigl(x\cdot c_t-t\abs x^2/2\bigr)$ . Bayes’ rule for conditional laws therefore identifies (D17.3), for each fixed $t\ge0$ , as a version of the conditional law $\mathcal L(X\mid\mathcal F_t)$ :

\E_t\phi=\E[\phi(X)\mid\mathcal F_t]\quad\text{a.s.}

(D17.9)

for every $\phi\in L^1(\mu)$ (first for bounded $\phi$ , then by monotone convergence for nonnegative integrable $\phi$ , then by linearity). This is the same convention as in the certified dossiers for Lemma 21.2 and Lemma 21.1.

The posterior potential $V(x)-c_t\cdot x+\tfrac t2\abs x^2$ has Hessian $\succeq(\varepsilon+t)I_n\succ0$ , so $\mu_t$ is a smooth strongly log-concave probability measure with everywhere positive density; in particular it has all polynomial moments and its covariance $A_t$ is positive definite (a nontrivial linear functional cannot be $\mu_t$ -a.s. constant when the density is positive on all of $\R^n$ ).

All posterior moments in this proof are jointly measurable in $(t,\omega)$ : by (D17.3) they are explicit ratios of integrals of jointly measurable integrands against the fixed measure $\mu$ , evaluated at the jointly measurable pair $(t,c_t(\omega))$ .

Step 1: the tensor is defined off a null set. Fix $t\ge0$ . By (D17.9) applied to $f^2$ and the tower property, $\E[\E_tf^2]=\E_\mu f^2<\infty$ , so $\E_tf^2<\infty$ almost surely. By Tonelli’s theorem the set of pairs $(t,\omega)$ with $\E_tf^2=\infty$ is $\dd t\otimes\dd\Prob$ -null. Off this null set, conditional Cauchy–Schwarz gives

\E_t\bigl[\abs{f-m_t}\,\abs{X-a_t}^2\bigr] \le v_t^{1/2}\,\bigl(\E_t\abs{X-a_t}^4\bigr)^{1/2}<\infty,

(D17.10)

because $\mu_t$ has all moments. Hence $H_t$ is defined by an absolutely convergent integral and is a symmetric matrix, for almost every $(t,\omega)$ .

Step 2: pathwise whitened duality. Work at a fixed pair $(t,\omega)$ off the null set of Step 1. Since $A_t\succ0$ , the whitened variable

Y=A_t^{-1/2}(X-a_t), \qquad X\sim\mu_t,

(D17.11)

is well defined; its law is an affine image of the log-concave law $\mu_t$ , hence log-concave, and it is centered with covariance $A_t^{-1/2}A_tA_t^{-1/2}=I_n$ : it is isotropic log-concave. Write $\widehat H_t=A_t^{-1/2}H_tA_t^{-1/2}$ . For any symmetric matrix $M$ ,

\begin{aligned} \inner{M}{\widehat H_t}_{\HS} &=\E_t\bigl[(f-m_t)\,(X-a_t)^TA_t^{-1/2}MA_t^{-1/2}(X-a_t)\bigr]\\ &=\E_t\bigl[(f-m_t)\,Y^TMY\bigr] =\Cov_{\mu_t}\bigl(f,\,Y^TMY\bigr), \end{aligned}

(D17.12)

the last equality because the first factor is centered. The quadratic polynomial $Y^TMY$ lies in $L^2(\mu_t)$ , so conditional Cauchy–Schwarz and Theorem 25.1 applied to the isotropic log-concave law of $Y$ give

\inner{M}{\widehat H_t}_{\HS} \le v_t^{1/2}\,\Var_{\mu_t}\bigl(Y^TMY\bigr)^{1/2} \le v_t^{1/2}\,\sqrt8\,\norm M_{\HS}.

(D17.13)

Since $\widehat H_t$ is symmetric, taking the supremum over symmetric $M$ with $\norm M_{\HS}\le1$ recovers its full Hilbert–Schmidt norm and yields (D17.8):

\norm{\widehat H_t}_{\HS}^2\le8v_t .

(D17.14)

This is a pathwise, per-time inequality; no expectation, no stopping, and no independence statement has been used.

Step 3: unwhitening strictly before the exit time. On the event $\{t<\tau_L\}$ the definition of the infimum gives $\norm{A_t}_\op<L$ . Writing $H_t=A_t^{1/2}\widehat H_tA_t^{1/2}$ and using the ideal property of the Hilbert–Schmidt norm,

\norm{H_t}_{\HS} \le\norm{A_t^{1/2}}_\op^2\,\norm{\widehat H_t}_{\HS} =\norm{A_t}_\op\,\norm{\widehat H_t}_{\HS},

(D17.15)

so that, combining with Step 2, for almost every $(t,\omega)$ ,

\norm{H_t}_{\HS}^2\,\one_{\{t<\tau_L\}}\ \le\ 8L^2\,v_t .

(D17.16)

The operator norm was used only where it is deterministically below $L$ ; no moment of $\norm{A_t}_\op$ appears.

Step 4: the variance budget and integration. Fix $t\ge0$ . By (D17.9), $m_t=\E[f(X)\mid\mathcal F_t]$ a.s., so $\E m_t=\E_\mu f$ and, by Jensen’s inequality, $\E m_t^2\ge(\E_\mu f)^2$ . Hence

\E v_t=\E\bigl[\E_tf^2-m_t^2\bigr] =\E_\mu f^2-\E m_t^2 \le\E_\mu f^2-(\E_\mu f)^2 =\Var_\mu(f)=1 .

(D17.17)

Finally, since $\one_{\{t<T\wedge\tau_L\}}\le\one_{\{t<\tau_L\}}\one_{\{t<T\}}$ , Tonelli’s theorem for the nonnegative integrand, (D17.16), and (D17.17) give

\E\int_0^{T\wedge\tau_L}\norm{H_t}_{\HS}^2\dd t \le\int_0^T\E\bigl[\norm{H_t}_{\HS}^2\one_{\{t<\tau_L\}}\bigr]\dd t \le8L^2\int_0^T\E v_t\dd t \le8L^2\,T .

(D17.18)

This proves (D17.7).

Remark D17.1 (What is, and is not, claimed)

The bound (D17.7) is confined to times strictly before the covariance exit time $\tau_L$ ; it makes no assertion about $\E\int_0^T\norm{H_t}_{\HS}^2\dd t$ without the stop, no universal-time occupation claim, and no claim about Conjecture 21.1 itself. The fenced product $\E\bigl[v_t\norm{A_t}_\op^2\bigr]$ is never formed: the operator norm enters only through the deterministic pathwise bound of Step 3, so the recorded marginal-probability independence fallacy is not engaged. By Proposition 0.1, the event $\{\tau_2\le t\}$ ceases to be rare past times of order $1/\log n$ , so this mechanism cannot be extended to universal time; that limitation is inherited, not violated, here.

Obstructions respected. The node carries no bounded_by edge; the registered route fences were checked individually. No cut, slice, or excess estimate occurs (rem:two-tail-slice-bounds, rem:profile-circularity, rem:single-coordinate-cuts). The only tensor input is the full symmetric-matrix quadratic-chaos bound of Theorem 25.1, used in its certified all-law form; no radial or projection test is promoted to a dimension-free chaos bound (rem:projection-ceiling). No crude covariance integral, no relative occupation bound, and no covariance bootstrap appears (rem:crude-insufficient, rem:relative-ceiling). The covariance-spike obstruction (Proposition 0.1) is respected constructively: the estimate stops at the exit time precisely because operator-norm control past the window is false.