Small-time operator-norm control of the covariance

Both variants of the fixed cut, the all-cut and the near-Cheeger variant, meter the danger of localization through the same cut-free covariance functional. Recall from (28.8) the covariance excess $X_t=(\lmax(A_t)-1)_+$ and the interface functional $\Xi_T(\mu)=\int_0^T\E X_t\dd t$; its second-moment companion is

The bootstrap of Chapter The fixed cut: the bootstrap uses $\Xi_T$ and the covariance reduction of Chapter The fixed cut: product stress test uses $\Xi^{(2)}_T$; both rest on small-time operator-norm control of the covariance. We isolate that control as an assumption and derive it from the literature: with exponent $C_2=2$ from the published sup-time estimate below (Corollary 26.2), and with $C_2=1$ under Letwin’s quadratic estimate Theorem 25.1 (Corollary 26.3).

This is the shape of the estimate underlying the Chen bootstrap as presented in Klartag’s lectures. The argument through the Cheeger bounds available before 2026 gives $C_2=2$. Letwin’s dimension-free quadratic Poincaré inequality (Theorem 25.1) instead bounds the third-moment parameter $\kappa_n$ universally; inserted into the Klartag–Lehec moment bound, which holds for $t\le(C\kappa_n^2\log n)^{-1}$, it gives $C_2=1$. The standard two-sided-exponential product example suggests that $1/\log n$ is the natural endpoint for covariance-only control: the largest time scale it can reach, since the top covariance eigenvalue of that product reaches order $\log n$ at times of order $1/\log n$.

This bound is what makes the small-gap implications of Section The initial layer: before and after a covariance exit vacuous. The argument for Theorem 4.6 sharpens the dimension dependence without using it; no covariance statement below gives control up to a universal time.

The parallel-coupling preprint contains rank-sensitive information that is stronger than early-time operator-norm control but still does not see a cut. The integrated rank estimate below uses the stopped rank estimate; the third-moment and covariance-moment arguments after it use separate inputs.

Both statements follow version 2 of Klartag & Lehec, 2025. They control how many covariance eigenvalues are large and how long each rank can remain large. They do not control the orientation of a cut tensor $K_t$, a posterior Hessian $H_t$, or an eigenfunction source relative to those eigenspaces. In particular, neither theorem by itself discharges Conjecture 29.1, Conjecture 21.1, or the high-rank part of Conjecture 29.3.