Values of polynomial continued fractions with Liouvillian Gauss equations

We study values of nonterminating polynomial continued fractions with a finite limit, with rational coefficients, a quadratic numerator and a linear denominator, whose Gauss hypergeometric equation is Liouvillian. We separate sectors: algebraic transfer when a contiguous shift is nondegenerate, incomplete-Beta representations in the reducible sector, elementary rational-exponent reductions, and separately qualified algebraic-curve and irrational-hyperexponential sectors. These are structural reductions under stated hypotheses, not a blanket membership theorem. Finite reference monodromy alone does not force an algebraic limit: an explicit example has the transcendental value −9√3/(3√3+2π) because the shift degenerates. Five first-kind limits are proved transcendental over the algebraic numbers using the Huber-Wustholz theorem. For a(n)=n²−2, b(n)=n+1/2 the limit is √2·cot(√2·arccos(1/√5)), transcendental by Gelfond-Schneider; conditional on Schanuel's conjecture it is transcendental over the one-level field generated by exponentials of algebraic numbers, although it is an elementary number in Chow's iterated sense (an EL number). A numerical consistency check finds all 95 recorded detections of an earlier census in the algebraic or elementary sectors, with one regularized record outside the ordinary domain; this is evidence, not proof. The general abelian-sector conclusions keep their stated hypotheses, and the original non-Liouvillian conjecture remains conjectural. Preprint, not peer reviewed. MSC2020: 11J81 (primary); 11J70, 33C05, 34M15. AI Declaration Tools. AI assistance for this paper was obtained through GitHub Copilot CLI (versions 1.0.92 to 1.0.97 as recorded in the session logs) and, for advice only, through the claude.ai chat interface (see Advisory model below). The models are named below by vendor label and by the model string recorded in those logs. OpenAI GPT-6.1 Sol (gpt-6.1-sol). Constructed the mathematical arguments of this paper from the author's task statements: the sector classification, the degenerate-shift example, the applications of the Huber–Wustholz theorem, the irrational dihedral family and the Schanuel argument, including the written Gelfond–Schneider and Schanuel proofs, following the route suggested by the advisory model. It wrote and ran the exact-arithmetic and census recount scripts, drafted the sections of the paper and the outline from which they were drawn, and read the built PDF in a reading check. Anthropic Claude Sonnet 5.5 (claude-sonnet-5.5). Reviewed the mathematical arguments and the drafted sections as a single machine reviewer per round (cross-vendor review of the sector theorem, review of claim preservation and framing, review of PDF formula fidelity). It also converted the Markdown draft to the LaTeX source, ran the build, inserted the Supplement S archive reference into the submission copy, and wrote this declaration and the next section. Anthropic Claude Sonnet 5 (claude-sonnet-5). Reviewed the draft of section 2 in one round. The review record names only Anthropic Claude, without a variant; the model string claude-sonnet-5 is taken from the session log. Anthropic Claude Opus 5.5 (claude-opus-5.5; session metadata records claude-opus-5) and OpenAI GPT-5.6 Sol (gpt-5.6-sol). The H2 experiment whose recorded results section 3 recounts was implemented and run by agents using claude-opus-5.5; its preregistration was reviewed by agents using gpt-5.6-sol. Neither contributed to the proofs. Advisory model. Claude Opus 5.5 (Anthropic; model selection not logged), used through the claude.ai chat interface, gave strategic and mathematical advice: it proposed the second paper and its outline, identified the Gelfond-Schneider structure of the value for a(n) = n² - 2 and the Schanuel route to Theorem 2.3, suggested the exact-arithmetic re-derivation, and reviewed reports of the work. The degenerate-shift counterexample (Example 1.2) was found in a GitHub Copilot CLI session. Extent of contribution. Mathematical argument construction, calculations and the wording of the text were produced largely by the models above. Research directions were set by the author. Several specific ideas came from the models, as described above; the proofs were written by the models, not by the author. What was checked by whom. The author checked by hand the Gauss parameters and the Kimura classification of the dihedral family of Proposition 2.2, and compared its closed form with a hand evaluation of the continued fraction tail; these checks did not cover the Gelfond–Schneider argument or the Schanuel-conditional argument. All other proofs, namely Theorem 1.1, Example 1.2, Proposition 2.1, the transcendence arguments of Proposition 2.2 and Theorem 2.3, and the census recount of section 3, were checked only by AI review (the reviewers named above, the authoring model's own checks and scripted exact-arithmetic checks). AI review is not independent human validation and is not represented as such. Responsibility. The author takes full responsibility for the content of this paper. Supplementary records: https://doi.org/10.5281/zenodo.23313637 (version 1.0, as cited in the paper; all versions: 10.5281/zenodo.23313636)