Identifiability and calibration limits of a single dispersion branch in a two-field effective theory

We study what one dispersion branch can determine about a two-field quadratic effective theory, and which additional assumptions are required to turn exact reconstruction into a reliable inference from data. In a fixed metric and field normalization, three distinct positive-momentum samples determine the three pole parameters away from an exceptional linear branch. We characterize the complete exceptional family and show how a calibrated response residue resolves it. Exact finite-error identities, including jointly uncertain momentum and frequency, connect this structural result to conditional error certificates. An explicit classical Gaussian heavy-field extension supplies a checkable matching example, while also showing why positivity alone does not bound the displacement of a heavy pole. We then examine published plasmon-polariton data through the separate zero-gradient boundary of the model. The full pilot contains 167 peak positions in five particle-size groups; a subsequent layer-resolved comparison uses five interior points with matched endpoint fitting. Its energy RMSE is 20.89 meV, compared with 25.79 meV for a two-parameter saturation curve, without a statistical significance claim. We derive an exact family under unknown common momentum calibration and certify nominal energy-tolerance floors for four layer groups. An audit of the archived spectra identifies missing uncertainty information and limits on reproducing the original peak extraction. The result is a conditional inverse framework and a reproducible empirical audit, not an independently certified measurement of the unobserved upper gap.