Spectral Possibilistic Identification: Sharp Misfit–Spectral Bounds and Data-Compatibility Stability Margins

An accurate fit to a finite response record may leave both stable and non-Hurwitz models compatible with the observations. Spectral Possibilistic Identification (SPI) measures the additional observation loss required to admit a non-Hurwitz model within a prescribed parameter family and identifies strictly stable possibility cuts. For a cyclic Jordan family of dimension n, explicit response bounds and the spectrum yield the sharp misfit–spectral exponent 1/(2n). The minimum non-Hurwitz loss occurs at the stability boundary and scales as Θ(β2n) as nominal damping decreases. A mass–spring–damper realization illustrates the theory, while nonlinear cart–pole simulations examine observation duration, outputs, and excitation. For the original six-second cart–pole experiment, the local quadratic approximation underestimates the full-loss margin by 82.13%. Lohner-type validated integration encloses the ideal-data margin in [0.93777,0.93778]. Exact spectral profiling has the same optimization target as SPI; changing the output direction can reverse the quadratic approximation error. The margin can help assess whether an allowed fitting-loss increase excludes non-Hurwitz alternatives or whether further observations are needed. Its interpretation depends on the model family, residual normalization, and experiment. The sharp law concerns the specified defective scalar family; the nonlinear certificate establishes local Jacobian stability for the ideal-data benchmark, not global stability or a noisy-data guarantee.