The Statistical Value of Resolution in Heavy-Tailed Dependent Residual Fields

Structured predictors often return a residual field with many coordinates per observation, but higher resolution need not provide proportionally more statistical evidence. We ask a basic question: under heavy tails and structured dependence, when does an additional residual coordinate actually reduce uncertainty? For a valid dependency graph on the residual field and a finite centered $q$-th moment, $q\in (1,2]$, we derive a profile-aware evidence count $\text{Neff}(\text{cC})=(nm/{\sum }_{\ell }{N}_{\ell }^{1/q}{)}^{q/(q-1)}$ that controls generalized Catoni risk estimation. Fractional independent-set covers give a graph-level refinement that can increase the evidence count, subject to support-size and confidence costs, while constant-probability lower bounds on canonical coordinate and space–time block families show that the dependence penalty is unavoidable up to logarithmic factors. We expose the graph assumption through safe supergraphs, independent-pilot transfer, conservative complete-graph fallbacks, and approximate-independence extensions. Experiments support the predicted finite-$q$ scaling trend, show severe undercoverage when the dependence/color factor is undercounted, and reveal a $10.49\times$ average uncertainty inflation on bounded CIFAR-10 residual maps. Decision audits further show that the value of dependence information is target-specific: it can change uncertainty dramatically without changing point rankings, and frozen dependence-aware risk-control rules can still abstain at small sample sizes. The resulting framework separates observed resolution from decision-relevant evidence.