The fixed eigenfunction: following one eigenfunction through localization

Overview of the mechanism¶

The idea. Follow a fixed first eigenfunction, rather than a cut, through stochastic localization, and apply the moment-map quadratic control of Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels to its whitened posterior covariance tensor, so that the eigenfunction’s tensor orientation survives in the posterior covariance. The target is an absorptive, function-aware estimate of source against damping over a universal amount of localization time. The mechanism works directly on the spectral object that defines the Poincaré constant, and keeps the orientation that a global operator-norm bound discards (see What fails, and why below).

What it would add. A localization mechanism that ignores covariance spikes: the object followed is one eigenfunction, which does not see spikes in directions it does not use, so the obstruction Proposition 0.1 does not apply to it. The mechanism reaches the Poincaré bound through one implication, Proposition 21.1, whose hypothesis is the occupation estimate Conjecture 21.1:

\text{fixed-eigenfunction full-damping occupation} \ \Longrightarrow\ \text{KLS}.

(21.1)

In words: as localization proceeds, the eigenfunction feeds a source into its covariance with the coordinates and the covariance of the localized measure damps it. The occupation is the source accumulated over time, $\E\int_0^t\norm{H_s}_{\HS}^2\dd s$ , and full damping means that the estimate may charge it against the whole damping term, with coefficient one, rather than against a fraction of it as the fixed cut must. The implication is proved; its hypothesis Conjecture 21.1 is not settled here.

What it builds on. From the literature: the quadratic input Theorem 25.1 and its directional third-moment consequence Proposition 26.1, and the published early-time covariance bound Theorem 26.1. Developed here: the exact fixed-function SDE and the posterior-defect calculus (Lemma 21.1).

What it gives. The stopped source estimate Lemma 21.3 controls the accumulated source before a covariance exit, using Theorem 25.1. Together with the time-weighted source and restart estimates, it describes how a post-exit charge could be bounded.

What blocks it. Conjecture 21.1 asks for the full-damping source estimate on a universal time interval, uniformly on regular approximants, without losing tensor/covariance alignment under unwhitening. The stopped estimate alone does not control the source after covariance exits. Nor does the small-gap implication Proposition 21.2, whose hypothesis no measure satisfies (Section The initial layer: before and after a covariance exit).

What fails, and why. No variant of this mechanism is known to fail, which reflects how little it has been explored rather than its strength. The binding constraint is external: Proposition 0.1 forbids the uniform operator-norm bound that a coarser version of this argument would want, which is precisely why the mechanism keeps the eigenfunction’s tensor rather than the covariance’s top eigenvalue.

What would settle it. Conjecture 21.1 gives KLS through Proposition 21.1; a family of measures on which unwhitening necessarily loses alignment would test the limits of this mechanism. The stopped-source argument uses Theorem 25.1; Section Scope of the quadratic input describes the scope of that input.

How to read it. Conceptual prelude: Chapter Prelude: what stochastic localization does, and what it costs — this mechanism shares it entirely with the fixed cut and differs only in the object followed. Moment-map control it uses: Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels. Apparatus: the shared technical foundations, from Chapter Analytic conventions and the two-color localization setup on.

Setup and the exact fixed-function SDE¶

Work first on smooth, strongly log-concave isotropic approximants, where the weighted Laplacian $L=\Delta-\nabla V\cdot\nabla$ has discrete spectrum. Let $f$ be a normalized first nonconstant eigenfunction, $-Lf=\lambda f$ , and define along the localization of Section The process

g_t=\Cov_{\mu_t}(f,X), \qquad H_t=\E_t\bigl[(f-\E_tf)(X-a_t)^{\otimes2}\bigr], \qquad A_t=\Cov_{\mu_t}(X).

(21.2)

Here $g_t\in\R^n$ is the covariance of $f$ with the coordinates, and $H_t$ is the symmetric-matrix-valued tensor coupling $f$ to the second-order posterior structure.

The exact evolution is

\dd g_t=H_t\dd W_t-A_tg_t\dd t .

(21.3)

So $\norm{H_t}_{\HS}^2$ is the source for $\abs{g_t}^2$ and $2g_t^TA_tg_t$ is its exact damping. This is the fixed-function analogue of the two-color Riccati identity of Chapter The two-color Riccati identities: source and damping are separated exactly, with no inequality spent.

What the July 2026 input gives, and what it leaves¶

By Theorem 25.1, the eigenfunction tensor satisfies the optimal intrinsic static estimate in whitened coordinates,

\norm{A_t^{-1/2}H_tA_t^{-1/2}}_{\HS}^2\le8\,\Var_{\mu_t}(f).

(21.4)

This is the exact analogue, for the fixed-eigenfunction object, of the static quadratic-chaos input that Chapter The static quadratic-chaos input and the two-tail obstruction supplies for the fixed-cut object.

Crude unwhitening of (21.4) — multiplying back by $A_t^{1/2}$ on both sides — leaves a factor $\lmax(A_t)^2$ , and hence exactly the universal-time dynamic alignment problem that Section Where the surviving logarithm lives identified as the residual cost. So the quadratic estimate does not remove the difficulty; it relocates it to an object with more structure.

Lemma 21.1 (Posterior eigenfunction-defect calculus — Proved · agent review (2026-08-27) · lem:mm-posterior-defect · Correction)

In the regular setting above, realize localization through the planted channel $c_t=tX+B_t^{\mathrm{obs}}$ , where $X\sim\mu$ and $B^{\mathrm{obs}}$ is an independent Brownian motion, and put

R_t(x)=(c_t-tx)\cdot\nabla f(x), \qquad \rho_t=R_t-\E_tR_t, \qquad m_t=\E_tf.

(21.6)

Define

b_t=\E_t\nabla f, \quad C_t=\E_t[(X-a_t)\otimes\nabla f], \quad u_t=\E_t[\rho_t(X-a_t)], \quad K_t=\E_t[\rho_t(X-a_t)^{\otimes2}].

(21.7)

Then, for every $t\ge0$ ,

\begin{aligned} \E_tR_t&=\lambda m_t, &\lambda g_t&=b_t+u_t, &\lambda H_t&=2\operatorname{sym}C_t+K_t, \end{aligned}

(21.8)

\begin{aligned} \E\Var_t(R_t) &=t\lambda-\lambda^2\int_0^t\E\abs{g_s}^2\dd s, &\E\int_0^t\norm{C_s}_{\HS}^2\dd s &\le\lambda-\lambda^2\abs{g_0}^2, \end{aligned}

(21.9)

\begin{aligned} \E\E_t\abs{\nabla R_t}^2 &\le t\lambda^2+t^2\lambda. \end{aligned}

(21.10)

These identities do not by themselves unwhiten the high-covariance part of $H_t$ .

The central problem¶

Proof 21.1

For a normalized first eigenfunction in the regular class, the fixed-function SDE (21.3) and Itô’s formula give

q(t)=\abs{g_0}^2+ \E\int_0^t\bigl(\norm{H_s}_{\HS}^2-2g_s^TA_sg_s\bigr)\dd s, \qquad q(t)=\E\abs{g_t}^2.

(21.13)

Bessel’s inequality gives $\abs{g_0}^2\le1$ . The full-damping occupation hypothesis cancels the entire last term and Grönwall yields $q(t)\le M_*$ on $[0,T_0]$ . Since

\E\Var_{\mu_t}(f)=1-\int_0^tq(s)\dd s,

(21.14)

the left side is at least $1/2$ at $t=T_*$ . Posterior Brascamp–Lieb and the fixed-test tower property bound it above by $T_*^{-1}\E\abs{\nabla f}^2=\lambda/T_*$ , so the first eigenvalue satisfies $\lambda\ge T_*/2$ . Smooth strongly convex approximants, whitening, and passage of the uniform Poincaré inequality on fixed smooth tests give the same bound for every isotropic log-concave law, without convergence of eigenfunctions. The full domain and approximation argument is in the full proof linked from the statement.

The form of (21.11) is worth emphasizing. Here the source may be charged against the full exact damping. In the evolution of $\E|g_t|^2$ those two terms then cancel, leaving a closed Grönwall inequality from the linear budget and the lower-order term. A strict damping surplus would be useful but is not required for the KLS sufficiency bridge; this differs from the absorption margins required in the two-color setting of the fixed cut.

Lemma 21.2 (Time-weighted fixed-function source budget — Proved · agent review (2026-08-27) · lem:mm-time-weighted-fixed-source · Correction)

Let $\mu=e^{-V}\dd x$ be a smooth probability with $\Hess V\succeq\kappa I$ , $\kappa\ge0$ , and let $f$ be a fixed square-integrable test for which the posterior identities are justified. Put $g_t=\Cov_{\mu_t}(f,X)$ , $H_t=\E_t[(f-\E_tf)(X-a_t)^{\otimes2}]$ , and $A_t=\Cov_{\mu_t}(X)$ . Then, for every $T>0$ ,

\begin{aligned} &\E\int_0^T(\kappa+t)\norm{H_t}_{\HS}^2\dd t +2\E\int_0^T\left(\abs{g_t}^2-(\kappa+t)g_t^TA_tg_t\right)\dd t \\ &\hspace{7em}\le \Var_\mu(f)-\kappa\abs{g_0}^2. \end{aligned}

(21.15)

In particular, for $\kappa=0$ , $\E\int_0^\infty t\norm{H_t}_{\HS}^2\dd t\le\Var_\mu(f)$ . This estimate retains one power of time and by itself gives no unweighted initial-layer bound.

The initial layer: before and after a covariance exit¶

The next statements separate control before a covariance exit from the cost of restarting afterwards. The stopped source uses Theorem 25.1 and no further hypothesis. The last two statements assume a small gap, $\lambda\le3/(8K_n)$ , and no measure satisfies it: Theorem 26.2 gives $\lambda=1/\CP(\mu)\ge1/K_n>3/(8K_n)$ . Those two implications are therefore vacuous; they supply no time interval on which Conjecture 21.1 holds for an actual measure, and the bound $\CP\le C\log^2n$ they reproduce is weaker than the published one.

The mechanism is quadratic duality at each posterior. Center and whiten the posterior, pair its source tensor with a fixed symmetric matrix, and apply Theorem 25.1 followed by Cauchy–Schwarz. Hilbert–Schmidt duality gives (21.4). Returning to the original coordinates costs at most $\norm{A_t}_{\op}^2$ , hence at most $L^2$ before the exit. Finally, total variance gives $\E\Var_{\mu_t}(f)\le1$ ; integrating yields $8L^2T$ . This argument needs a fixed square-integrable test, not its eigenfunction equation.

The stopped estimate Lemma 21.3, unlike the last two statements, applies to every measure (see the beginning of this section).

An equivalent sufficient reformulation¶

There is a second, coarser way to state what this approach needs. Combining Proposition 26.1 with its quadratic input Theorem 25.1 gives $\kappa_n\le2\sqrt2$ . Hence any dimension-free comparison of the form

\CP(\mu)\le C\bigl(1+\kappa_n^2\bigr)

(21.17)

valid for every isotropic log-concave $\mu$ would prove KLS. Equivalently: it suffices to remove the residual $\sqrt{\log n}$ from the current spectral comparison (0.14) while retaining only a universal function of $\kappa_n$ . Formulation (21.17) is useful as a target statement but supplies no mechanism; Conjecture 21.1 is the mechanism-bearing form.

The $H^{-1}$ endpoint¶

A natural first lemma for this approach is an $H^{-1}$ residual bound. It is worth recording why it is not one: it is an endpoint rather than a step — a statement of the same strength as KLS, usable as a final target but not as a preliminary lemma, although it looks like one.

With $b=\int\nabla f\dd\mu$ , the proposed target was

R(f)=\sum_i\norm{\partial_if-b_i}_{H^{-1}(\mu)}^2\le C\lambda .

(21.18)

It is still sufficient for KLS. But it is also quantitatively implied by KLS: one has the two-sided relation

1+\abs b^2-\frac{2\abs b^2}\lambda\ \le\ R(f)\ \le\ 1-\frac{\abs b^2}\lambda .

(21.19)

Its exact heat representation controls the short-time and high-frequency parts at the desired scale; the long-time low-spectrum tail carries the full strength of the Poincaré bound. Accordingly (21.18) is retained as a calibrated endpoint and diagnostic, not advertised as a likely preliminary lemma.

A second natural first attempt fails outright: a direct unweighting of the positive moment-map Stein form is false on truncated-exponential first eigenfunctions. This is why (21.4) is stated in whitened form.

Which obstructions apply here, and comparison with the other approaches¶

The obstructions proved elsewhere in this document are scoped, and it matters which of them bind here.

The projection ceiling of Chapter The static quadratic-chaos input and the two-tail obstruction does not obstruct this approach: the moment map uses information beyond radial and projection tests.
The single-coordinate product-budget result of Chapter The fixed cut: product stress test (Corollary 31.1) is specific to a fixed-cut localization counterexample and imposes no no-go here.
A universal Lipschitz Gaussian transport is too strong for exponential tails (No pathwise dimension-free Lipschitz map exists); the weaker expected-Jacobian criterion (5.1) is sufficient for KLS but currently reuses KLS-scale input, so it ranks below the target of this chapter.
Classical needles do not preserve the isotropic covariance constraints (Chapter Family 1: classical needle localization); that is a structural warning, not a theorem excluding all needle arguments.