Layerwise Terminal Discrepancy in Chen's Reverse-Heat Coupling on the Boolean Cube

Abstract

Recently, Chen \cite{Chen2026} proved that Talagrand's Boolean convolution conjecture holds up to the dimension-free factor \((\log\logη)^{3/2}\), namely for every fixed \(τ>0\), \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{(\log\logη)^{3/2}}{η\sqrt{\logη}}, \qquad η>e^3. \] We revisit the terminal testing-discrepancy step in Chen's perturbed reverse-heat coupling. Chen estimates this discrepancy globally in terms of the remaining gap to the terminal level. We keep the same coupling and the same reverse-heat formulations, but localize the terminal discrepancy on each remaining-gap layer before summing the layers. This changes the fixed-time anti-concentration cost from order \((\log L)^{3/2}/\sqrt L\) to order \((\log L)/\sqrt L\), where \(L=\logη\). Consequently, we obtain a \((\log\logη)^{1/2}\) improvement as \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3. \]