Linear-denominator families of a census of polynomial continued fractions: Nörlund matching and a route classification

This note records a computational study 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: 46,656 families, of which 23,616 converge by the census rule. No family with a negative discriminant converges. What was done. (1) Under tail shift, sign change and constant scaling the converging families fall into 9,064 classes (the member lists of 1,277 classes were truncated in the enumeration, so their member counts are lower bounds). The three maps are stated in 16 theorems (with their sub-parts) in Lean 4 and proved by the Lean kernel; 3,297 relations of the census were checked against the lemmas. No human has read the proofs. (2) The formula of Nörlund, as quoted by Campbell and Chen, was evaluated for one representative of each class and compared with the numerically computed limit: 8,651 agree to at least 157.3 digits, none disagrees, and the other statuses (formula undefined, infinite value, limit not obtained) are listed. (3) A pre-registered classification against a frozen list of 99 routes (85 active) from the cited literature gives each of 9,697 records (9,064 representatives and 633 supplementary members) a label by the pipeline from pinned sources: 429 EXPLAINED, 7,404 MATCH-ONLY, and the rest with a weaker kind of label or with no finite value. (4) A second pre-registered stage composes one quadratic transformation with a listed route; it adds 1 EXPLAINED record and no value that was not already known. (5) A cross-match with Cohen's database of continued fractions (version 5) finds 67 families with an entry (54 of constant denominator, 10 of the main set and 3 of the supplementary set), 26 distinct entries; for 65 pairs the value of the entry agrees with our limit to at least 79.8 digits, and no label is changed. What is not claimed. The note states no theorem that is new. MATCH-ONLY means that no listed route applies; it does not mean that the limit has no closed form. Every label is a label of a pipeline, from pinned sources: 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. Routes taken from the author's own preprint on Liouvillian Gauss equations carry the tier "author's preprint, AI-checked". The 7 ledgers (105 entries) record every stop and every amendment with its timing relative to the population results. Files. main.pdf is the note. lindenom_note_archive.zip holds the paper, the scripts, the staged stage folders (with local paths replaced by placeholders and a map of original and shipped hashes), the numbers generator with 250 assertions, and the checks. Sources, third-party text and the Cohen TeX are not shipped, only hashes, versions and locations; prose passages of third-party sources were removed from the staged files and replaced by a location and a hash (the list is in the archive). The archive regenerates the numbers, the paper and the checks and verifies the ledgers; 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) and the preprint on Liouvillian Gauss equations (concept 10.5281/zenodo.23357761). Text and PDF: CC BY 4.0. Code: MIT.