Two-threshold identification and sharp held-out bounds in a positive spectral model

We study a once-subtracted vacuum-polarization model in which positive mixtures of the massive Dirac one-loop profile are observed on a uniform spacelike grid in the presence of an unknown quadratic background. The lightest allowed threshold, the scale, and the value at the origin are fixed. Five samples at positive grid points uniquely identify a two-threshold mixture with distinct thresholds and positive weights, whereas four samples admit a continuum of fits. A three-threshold mixture can match the same five samples exactly. Over all positive mixtures with those samples, the two-threshold fit gives the smallest possible sixth value. The sharp upper bound is approached as one threshold tends to infinity and its weight grows. Finite differences remove the background and reduce the problem to a Chebyshev moment space. Two outward-rounded interval calculations certify the required signs for dimensionless grid steps up to one for identification and up to one half for the sharp bounds. The accompanying material contains the LaTeX source, synthetic inputs, code for both interval checks, and reference outputs.