One plane, three regimes: polynomial continued fractions with quadratic numerators, organised by Δ and the degree of the denominator

This is an expository survey of continued fractions with quadratic partial numerators and partial denominators of degree at most one. It sorts what is known, and what the author has checked, by the discriminant Δ = β1² + 4α2 and by the degree of the denominator. It claims no new theorem and is not peer reviewed. The three regimes (linear denominator with Δ ≠ 0; Δ = 0; constant denominator) are the author's own organisation, not a published taxonomy. Every statement carries a label that records its source and how strongly it is supported (CITED, CD for cited-derived, LEAN, PMS, VERIFIED, TMA, EXPL, HEUR, OPEN); nothing is stated more strongly than its source. LEAN means a statement proved in Lean 4 as stated, by the pipeline, with the statements read by AI only. VERIFIED means two independent numerical methods agree to at least 50 digits, by a pipeline from pinned sources; it is not a proof. The census labels EXPLAINED, MATCH-ONLY and "no applicable route" are cited from census v1.2 (10.5281/zenodo.23764797) and mean "by that census pipeline", not proved. No human expert has refereed any part of the note or of the deposits it summarises. Version 1.1 adds three things from later work and no new theorem. (1) Theorem C for every real b > 0 and s = ±1: a Lean 4 development, shipped as lean_thmCb in the archive, proves that the continued fraction with partial numerators n² and partial denominators bs has non-zero denominators and converges to s/(b J(b)), with J(b) an explicit integral, together with the series form of b J(b); it states convergence and the limit only, with no rate and no bound, and the case b = 4 gives the conclusions of the earlier Theorem C. (2) The explanation, by named hypergeometric evaluations, of the 56 Trans families of the census, cited from census v1.2. (3) The classification of the 2,210 constant-denominator families into 1,276 EXPLAINED, 900 MATCH-ONLY and 34 with no applicable route, cited from census v1.2 and not recomputed. The archive plane_note_artifact.zip holds the three pieces of evidence the note refers to. (1) lean_thmCb: Lean 4 sources (Lean v4.30.0, Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f) of Theorem C for every real b > 0, four theorems; recompiled on 2026-10-30 with no sorry and the standard axioms only. (2) lean_bridge: Lean 4 sources (same versions) of a bridge linking the convergence statements for the boundary family Δ = 0 to the factorial-reduction statements of the Theorem B note, for members with rational roots and without termination; recompiled on 2026-10-29 with no sorry and the standard axioms. (3) census_recompute: a 60-digit recompute of the 2,176 constant-denominator census families against Nörlund's formula, with the table, report, pre-registration (a redacted copy: local paths and closed-folder names replaced by placeholders; the README gives the sha256 of the original) and scripts; the CF self-accuracy runs from 89.7 digits up to the method's 100-digit cap, and the 34 families covered only by a second formula are not upgraded. All were produced in operator-assist sessions with AI assistants, not with the registered instruments of the census project. The note is licensed CC BY 4.0; the code and Lean sources in the archive are licensed MIT.