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 It claims no new theorem and is not peer reviewed.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. No human expert has refereed any part of the note or of the deposits it summarises. The archive plane_note_artifact.zip holds the two pieces of evidence the note refers to. (1) lean_bridge: Lean 4 sources (Lean v4.30.0, Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f) 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. (2) 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. Both 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.