Rational moment algebras: spectral reconstruction, factorization and arithmetic rigidity

For rational characteristic functions with finite symmetric Laplace-mixture representations, simple poles and positive residues, we connect truncated moments to exact rational factorization data. Residue-weighted Hankel determinants recover a normalized numerator–denominator resultant, while equal spectral weights recover a polynomial discriminant. The associated finite rational spectral algebra is reconstructed from the Hankel pencil; its idempotents identify rational partitions of poles. For integral logarithmic-derivative selectors on real simple spectra, we prove a discriminant-quantized variance gap: the defect is either zero, corresponding to a genuine integral factor, or at least \(2/\Delta\) under the stated positivity hypotheses. Smith-normal-form denominators yield an exact local reconstruction threshold. We also give explicit limits of positive-moment inference: an everywhere locally soluble factor-degree problem can fail globally, and smoothly varying positive Fourier models may have densely alternating rational splitting types and discontinuous minimal contact radii. These results distinguish recoverability of a finite algebra from certification of its arithmetic factorization. No general-purpose factorization algorithm or claim about all positive rational characteristic functions is made.