Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation

We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity mathbb Esupx∈ TXx; 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 [Phiμ(x):=int04σ(x)√logfrac1μ(Bd(x,varepsilon)),dvarepsilon,] 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 EY~ Qπ(Bd(widehat t(Y),Delta)), 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 example separating the usual Fano relaxation, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope, and the full-class minimax risk. Together, these results show that algorithmic lower bounds provide local-geometric validations of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent. This separation can also be recast in minimax language as penalty-range information relaxation, highlighting an important question of algorithmic robustness for classical high-dimensional models and regularized algorithms.