Simultaneous Tolerance Certification: Limit Experiments and Sharp Resource Laws

Simultaneous tolerance certification asks whether every component of a dense latent vector lies within its prescribed range. This preprint studies noisy unit-norm linear measurements under pathwise resource budgets, with tolerances calibrated to a fixed prior membership probability. For independent Gaussian priors, every fixed linear channel is uniformly asymptotically equivalent, for the membership decision, to a coordinate-precision design at no greater total precision. A marked Poisson limit characterizes the complete family of feasible limiting experiments. The resulting allocation theory gives exact Brier-optimal designs and proves that every classification-optimal allocation has zero plus at most one positive normalized precision level. The global classification problem therefore reduces to a one-dimensional coverage choice. At low budgets, no allocation is most informative for every decision loss. At fixed row precision, adaptive screening yields a squared-logarithmic reduction in the order of measurements required for prescribed positive decision gain. For classification, sharp high-accuracy constants distinguish the inverse-square risk dependence of fixed designs from the logarithmic dependence of adaptive acquisition. Bounded-spectrum Gaussian dependence preserves the fixed-design frontier. Matching resource orders extend to exponential-power priors, including Laplace and lighter tails. A frozen Gaussian simulation, combined with certified deterministic bounds, establishes a finite-dimensional advantage over all fixed linear designs under the same affine acquisition budget, with one-sided 95% Monte Carlo confidence. This record contains the preprint, proof supplement, editable LaTeX sources, and a reproducibility archive with programs, experimental protocols, and saved results. The manuscript and proof supplement are licensed under CC BY-NC 4.0; original software is licensed under MIT. Third-party components retain their own licenses.