Conditional Model-Adequacy Tests for Spectral Uncertainty Claims in Lattice QCD
Euclidean lattice correlators determine spectral functions only through a smoothing integral transform, so a nominal uncertainty band on a reconstructed spectrum need not have a coverage interpretation for a physical summary. We formulate this as a target-wise adequacy test for reported spectral uncertainties. For a chosen summary \(T[ρ]\), the reported interval is tested on Euclidean-admissible mock correlators with known truth using empirical coverage, simulation-based calibration ranks, physical diagnostics, and stress tests. The test is conditional, but it is a useful falsification tool: passing it does not prove that a reconstruction is the QCD truth, while failing it shows that the reported uncertainty law is not adequate for the chosen functional under the stated mock extension. In a generic benchmark, peak locations are substantially better calibrated than peak heights or low-frequency weights, reflecting different degrees of functional identifiability under the Euclidean kernel. We then apply the same logic to a finite-temperature shear correlator. A family of BG-style reconstructions is compatible with the Euclidean data at \(χ^2/N_τ\simeq 1.3\). Within the scanned grid and stated observable-matched mock extension, a \(W_{\rm low}\)-calibrated representative can be identified, whereas pointwise peak-height intervals are not certified for the tested BG-style uncertainty law. Thus Euclidean compatibility is a necessary consistency check, but not a sufficient adequacy criterion for spectral uncertainty claims.