Rational moment algebras: spectral reconstruction, factorization and arithmetic rigidity

Abstract

We consider rational characteristic functions that admit finite symmetric Laplace-mixture representations with simple poles and positive residues. Such positivity does not by itself imply rational factorization; moment rank and rational splitting remain different properties. A general positive rational characteristic function need not belong to this class. Residue weights make the Hankel determinant measure a numerator–denominator resultant, whereas equal spectral weights recover a polynomial discriminant. Both observations reconstruct the same finite rational algebra, whose idempotents describe disjoint rational pole partitions. For integral logarithmic-derivative selectors on a real simple spectrum, we prove a sharp discriminant-quantized variance gap and a Smith-denominator bound that turns sufficiently accurate local data into an actual factor. The associated local covering can be exponential, as illustrated by Swinnerton–Dyer families. An everywhere locally soluble factor-degree layer can fail globally, as an explicit biquadratic example shows; a fixed-divisor obstruction alone cannot detect that failure. Finally, arithmetic specializations of a Chebyshev–Lucas pencil give smooth positive families with densely opposite rational splitting types and discontinuous minimum radii for rational Fourier contacts. These results distinguish exact algebraic reconstruction from robust arithmetic classification; they do not assert a new general-purpose factorization algorithm.