We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity $\mathbb E\sup_{x\in T}X_x$; the theorem here gives a simultaneous high-probability envelope for the entire field. For an ambient prior $μ$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[Φ_μ(x):=\int_0^{4σ(x)}\sqrt{\log\frac{1}{μ(B_d(x,\varepsilon))}}\,d\varepsilon,\] together with the corresponding Gaussian tail term. The theorem provides a reusable field-level refinement of classical generic chaining and a Gaussian-process counterpart of pointwise empirical-process bounds for deep neural networks. We also record a Bayesian algorithmic lower envelope from the interactive Fano/data-processing principle. For a known prior $π$, an observation channel, and a concrete estimator $\widehat t(Y)$, the lower bound is expressed through the exact ghost small-ball mass $\mathbb E_{Y\sim Q}π(B_d(\widehat t(Y),Δ))$, rather than a worst-case covering number. In Gaussian location experiments, comparison decoders convert Bayes location error into lower bounds on decision-aligned Gaussian ranges. We then construct an elementary weighted-basis example separating the usual Fano relaxation for a fixed prior, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope on the selected subatlas, and the full-class minimax risk/global Gaussian scale. Together, these results show that algorithmic lower bounds provide local-geometric certificates of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent.

翻译：我们证明了中心高斯过程的方差感知逐点主导测度定理。经典泛链式方法刻画标量量$\mathbb E\sup_{x\in T}X_x$；本文定理则给出整个场的同时高概率包络。对于背景先验$μ$，点$x$处的包络由逐点Fernique-Talagrand泛函 \[Φ_μ(x):=\int_0^{4σ(x)}\sqrt{\log\frac{1}{μ(B_d(x,\varepsilon))}}\,d\varepsilon\] 连同对应高斯尾部项共同决定。该定理提供了经典泛链式方法的可复用场级精化，以及深度神经网络逐点经验过程界的高斯过程对应版本。我们还基于交互式Fano/数据处理原理记录了贝叶斯算法下界。对于已知先验$π$、观测信道及具体估计量$\widehat t(Y)$，下界通过精确幽灵小球质量 $\mathbb E_{Y\sim Q}π(B_d(\widehat t(Y),Δ))$ 而非最坏情况覆盖数表达。在高斯位置实验中，比较解码器将贝叶斯位置误差转化为决策对齐高斯范围的下界。随后我们构造了基本加权基示例，分离了固定先验下的常规Fano松弛、贝叶斯算法下界、所选子图谱上的逐点高斯包络，以及全类极小极大风险/全局高斯尺度。这些结果共同表明：在过参数化背景类中，当经典极小极大理论变得过于粗糙或依赖预言机时，算法下界为固定估计量提供了局部几何化的逐点复杂度认证。