The parabolic (Δ = 0) families of a census of polynomial continued fractions: a Tricomi U formula and a route classification

This note records a computational study of the parabolic part (the discriminant Δ is zero) of the stratum of a census of polynomial continued fractions (see the census, version 1.3, 10.5281/zenodo.23899451) whose partial numerators are quadratic and whose partial denominators are linear, with integer coefficients between -4 and 4. That part has 2,916 families, of which 2,772 lie outside the family treated in an earlier preprint on a boundary family (concept 10.5281/zenodo.23581162); the families fall into 1,006 classes under tail shift, sign change and constant scaling. What was done. (1) Theorem D: when a further parameter θ is positive and the continued fraction does not terminate, its limit is b0 - λ(1 - ρ-)(1 - ρ+) U(a+1, c, θ) / U(a, c, θ), with U the Tricomi confluent hypergeometric function. The derivation was written by an AI model from statements of the literature that were read from the sources, with algebra checked symbolically in 22 cases, and was read a second time by another AI agent. No human has read it, and every value of this regime rests on it. (2) A pre-registered numerical match: of 331 classes addressed, 284 that do not terminate agree with the stated value to at least a preset number of digits (75, 25 and 40 digits by convergence type) and none disagrees; 47 terminate. 16 of the matched classes have θ = 0 (algebraic or logarithmic convergence) and lie outside the theorem: their limits were matched by extrapolation and no theorem is used. (3) A pre-registered classification of the 268 non-terminating classes of the theorem against a frozen list of three routes from the cited literature (a Bessel K function, an incomplete gamma function, an error function; special cases of U read from DLMF) gives 53 EXPLAINED and 215 MATCH-ONLY; the 47 terminating classes are counted apart as RATIONAL. (4) The 16 classes with θ = 0 and a limit (40 families) are binned NONCONVERGENT by the census rule although their convergents converge slowly; they are listed in a file of pending corrections and the census is not edited. What is not claimed. The note makes no novelty claim; the statements of the literature that it uses are listed by location only. MATCH-ONLY means that the limit and the formula agree and that no listed route applies; it does not mean that the limit has no closed form. For θ = 0 there is no theorem and for negative θ nothing is computed. Every label is a label of a pipeline, from pinned sources: the derivation, the routes, the transcriptions and the code were written and checked by AI models (Claude Sonnet 5.5, run as a research agent in GitHub Copilot CLI), and no human has refereed them. The handbook of Cuyt and others was not consulted. After the report of the last stage the operator re-tiered 57 route records from cited-derived to classical (a change of presentation; no label and no count changed), and a post-hoc remark on three classes was written; it is not a label and was not pre-registered. The ledger has 42 entries and records every stop and every amendment with its timing relative to the results. Files. main.pdf is the note. delta0_note_archive.zip holds the paper, the stage folders A to E (pre-registrations, outputs, scripts, the ledger and the briefs of the second readers; 123 files, with local paths replaced by placeholders and a map of original and shipped hashes), the numbers generator with 181 checks, and the checks of the paper. Sources and third-party text are not shipped, only hashes (367 rows), versions and locations; 5 prose passages of third-party sources were removed from the staged files and replaced by a location and a hash. The archive regenerates the numbers, the paper and the checks, verifies the ledger and reruns 11 stage scripts that read only shipped files; it does not rerun the stages from the sources. Companion records: the census (version 1.3, 10.5281/zenodo.23899451), the plane note (concept 10.5281/zenodo.23753323), the boundary-family preprint (concept 10.5281/zenodo.23581162), the note on the linear-denominator regime (version 1.1, 10.5281/zenodo.23998743; concept 10.5281/zenodo.23990753) and the preprint on Liouvillian Gauss equations (concept 10.5281/zenodo.23357761). Text and PDF: CC BY 4.0. Code: MIT.