Finite-Probe Total-Variation Certificates for Finite-Basis Drifting Models

• URL: http://arxiv.org/abs/2608.01547v1
• Date: Mon, 03 Aug 2026 00:00:40 GMT
• Title: Finite-Probe Total-Variation Certificates for Finite-Basis Drifting Models
• Abstract summary: We describe a frozen measurement system for integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis.<n>For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation.<n>We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degenera, and prove large-bandwidth collapse toward mean matching.
• Score: 1.2072720711193232
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator--denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.

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