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 \[ D_E\le C_τ\bigl(\cS_E+\sqrt{\cS_E\,\Pp(E)}\bigr), \qquad E\in\mathcal F_θ, \] where \(D_E\) is the localized terminal testing discrepancy and \(\cS_E\) is the stopped perturbative score energy. Applying this estimate to the layers \(G_r(θ)=\{r\le R_θ<r+1\}\) replaces the global Cauchy--Schwarz discrepancy cost by the layerwise cost \[ O_τ\left(\fracα{\sqrt r}+\frac{α^2}{r}\right) \Pp(G_r(θ)), \qquad α\simeq\log\logη. \] Under these imported joint-filtration inputs, combining the localized estimate with the time-smoothed anti-concentration profile yields the black-box consequence \[ μ\{P_τf>η\|f\|_1\} \le C_τ\frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3, \] for the Boolean heat semigroup. This makes a $(\log\logη)^{1/2}$ improvement over Chen's result.

翻译：我们在陈氏扰动逆向热方法~{\cite{Chen2026}}（针对布尔立方体上Talagrand卷积猜想）的终端检验差异步骤中，分离出一个逐层精细化处理。基于陈氏耦合的联合滤子鞅表述，以及下文联合滤子形式下可用的陈氏近似单调性和条件平方得分估计，我们证明了局部化检验估计： \[ D_E\le C_τ\bigl(\cS_E+\sqrt{\cS_E\,\Pp(E)}\bigr), \qquad E\in\mathcal F_θ, \] 其中\(D_E\)为局部化终端检验差异，\(\cS_E\)为停时扰动得分能量。将该估计应用于层\(G_r(θ)=\{r\le R_θ<r+1\}\)，可将全局Cauchy–Schwarz差异代价替换为逐层代价： \[ O_τ\left(\fracα{\sqrt r}+\frac{α^2}{r}\right) \Pp(G_r(θ)), \qquad α\simeq\log\logη. \] 在引入这些联合滤子输入条件下，将局部化估计与时间平滑反集中轮廓相结合，对布尔热半群可得黑箱推论： \[ μ\{P_τf>η\|f\|_1\} \le C_τ\frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3, \] 相较于陈氏结果实现了\((\log\logη)^{1/2}\)量级的改进。