Spectral Classes of Polynomial Continued Fractions: A Hypergeometric Framework and Arithmetic Obstructions

We propose a spectral classification program for polynomial continued fractions (PCFs) in which each family is assigned the invariant Spec (K) = (d, Λ, Δ, ρ, τ), recording the dominant degree, normalized Poincaré root, discriminant class, Wallis tail-decay rate, and arithmetic sector. This framework organizesclassical logarithmic and 1/π ladders together with higher-rank obstructedexamples inside a single reproducible taxonomy. Our first proved result is thePoincaré characteristic lemma, showing that balanced degree-one PCFs aregoverned at leading order by the quadratic law Λ² − αΛ − δ = 0. Our secondproved result is the Catalan logarithmic Frobenius obstruction lemma: afterfactorial normalization, a universal residual term 3/ (2n) forces a logarithmiccorrection and rules out the naive Gauss ₂F₁ ansatz. We then isolate two mainconjectures, one for a clean d = 2 Vquad regime linked to a confluent-Heun /Stokes problem and one for an Apéry-type d = 3 regime governed by aPicard–Fuchs operator and numerically identified with 6/ζ (3). The currentdataset contains 37 rigorously verified families, including seven rows importedfrom Cohen (2024) that supply elliptic and Gauss-hypergeometric comparisonexamples. The resulting picture suggests that arithmetic behavior in PCFs iscontrolled not only by coarse spectral data, but also by Frobenius resonance, monodromy type, and higher-rank geometry. Key open problems are to provenonreducibility in the clean obstructed sectors and to locate furtherresonant-versus-clean families. Companion archival deposit for a content-equivalent manuscript submitted to peer-reviewed venues; records of each submission differ only by author-block anonymization to satisfy each venue's blinding policy, while the mathematical substance is identical across versions. v1. 1 is the anonymized form per the host venue's double-blind requirements. For citation use the concept DOI 10. 5281/zenodo. 20173773, which always resolves to the latest version; see related-identifier entries for the upstream and lateral citation graph.