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

We isolate a layerwise refinement of the terminal testing-discrepancy step in Chen's perturbed reverse-heat approach~\cite{Chen2026} to Talagrand's convolution conjecture on the Boolean cube. Built on the joint-filtration martingale formulation of Chen's coupling, and on Chen's approximate monotonicity and conditional squared-score estimates being available in the joint-filtration form stated below, we prove the localized testing estimate \[ D_E\le C_\tau\bigl(\cS_E+\sqrt{\cS_E\,\Pp(E)}\bigr), \qquad E\in\mathcal F_\theta, \] where \(D_E\) is the localized terminal testing discrepancy and \(\cS_E\) is the stopped perturbative score energy. Applying this estimate to the layers \(G_r(\theta)=\{r\le R_\theta<r+1\}\) replaces the global Cauchy--Schwarz discrepancy cost by the layerwise cost \[ O_\tau\left(\frac{\alpha}{\sqrt r}+\frac{\alpha^2}{r}\right) \Pp(G_r(\theta)), \qquad \alpha\simeq\log\log\eta. \] Under these imported joint-filtration inputs, combining the localized estimate with the time-smoothed anti-concentration profile yields the black-box consequence \[ \mu\{P_\tau f>\eta\|f\|_1\} \le C_\tau \frac{\log\log\eta}{\eta\sqrt{\log\eta}}, \qquad \eta>e^3, \] for the Boolean heat semigroup. This makes a $(\log\log\eta)^{1/2}$ improvement over Chen's result.