Channel Theory for Polynomial Continued Fractions: Asymptotic Channels, the ξ₀ = 2/√β₂ Identity, and a Bridge Conjecture
papanokechi papanokechi · Zenodo (CERN European Organization for Nuclear Research) · 2026
Version notice (v1.7). This version corrects and attributes; it alters no theorem, proposition or numerical value of v1.6. Concept DOI is preserved (10.5281/zenodo.19941678); v1.6 (10.5281/zenodo.22782807) is superseded. It is a description-only version; the paragraph immediately below states what that means for the attached file. The file attached to v1.7 is the v1.6 build, unchanged. The v1.7 corrections are recorded in this description only. v1.7 is a description-only new version: no PDF was rebuilt, and none is claimed to have been. The v1.4–v1.6 LaTeX sources are not in the working repository, and the only source held is v1.3.1 — a build made from it would silently revert two versions of corrections, including the Painlevé-target correction made at v1.4 and the analytic residual added at v1.6, both of which this notice relies on. Attaching a build that cannot be verified is the v1.6 defect (1) repeated; not attaching one is the avoidable choice, and it is the one taken. If the v1.6 sources or PDF become available and an updated build is wanted, that is v1.8. 0. Current state This section is the whole current state. Nothing below it is needed to know what Channel Theory now asserts; §1 is history and §2–§7 are the detail. Proven. For Q_n = b(n)Q_{n−1} + Q_{n−2} with Q_{−1} = 0, Q_0 = 1, deg b = d ≥ 1 and b(k) ≠ 0 for k ≥ 1: — the normalised ratio r_n = Q_n/∏_{k≤n} b(k) converges unconditionally, with r_∞ − r_n = O(n^{−(2d−1)}); — if b(n) > 0 for all n ≥ 1 then 1 ≤ r_∞ ≤ exp Σ_{k≥1} 1/(b(k)b(k+1)) 0 (n ≥ 1), the identity ξ₀ = d/β_d^{1/d} was proved for the physical generating-function object by the holonomic/positivity argument of the companion EBR preprint 10.5281/zenodo.20564079 §3 (D-finite G; leading coefficient a_{2d}(s) = d^d s^{2d}(d^d − β_d s), roots {0, R}; Pringsheim), with scope positive-b families, physical object, symbolic + verified-to-d6 + Lean-checked algebraic layer d = 2–5, not a symbolic-degree formal proof. That superseded the Conjecture-3.3.A★ status for the positive-b case only. (2) Surface-type correction. All references — body, keywords and description — were corrected from Painlevé III(D₆) to the Painlevé V transcendent on the Sakai surface D₅^(1) (W(A₃^(1)), δ = −1/2 ≠ 0). Channel Theory was the last deposit carrying the stale P-III(D₆) root claim. v1.5 (2026-09-12) — a demotion. Clause (iii) of Theorem 3.7, the match between the CC-channel connection-coefficient ratio and the Riemann–Hilbert datum of P_V/D₅^(1) at the V_quad τ-parameter, is not a theorem of this paper and became the open problem op:cc-stokes-closed-form. Two grounds: the imported surface-type premise is itself conjectural (companion paper Conjecture 4.1, 10.5281/zenodo.20455089, read at v1.2), and the cited run stores a comparison its own verdict did not incorporate — a factor of about two accounted for by an explicit halving in the source, with ~1.7 % unexplained. v1.5 also withdrew as non-evidential a Domb–Sykes sentence that rounded two endpoints agreeing to two digits into a three-digit "range", and corrected five citations that pinned a version in which the cited object was not yet a conjecture. The v1.4 sentence "both corrections are additive/clarifying; no … claim is retracted" does not describe v1.5. v1.6 (2026-09-16) — five defects repaired; a build, labelling and open-problem erratum. (1) The v1.5 PDF was the draft build, watermarked "DRAFT — NOT DEPOSITED", its date line stalled at "2026-06-07 (v1.4, draft)" and its changelog ending at v1.3 — front matter, not text. (2) op:xi0-d3-direct withdrawn as vacuous, replaced by op:xi0-singularity-type, with rem:d3-vacuous recording the withdrawal (see §5). (3) A citation gap: 10.5281/zenodo.20542162 (deposited 2026-06-04) derives the slope-1/d edge polynomial χ_d(c) = 1 + (−1)^{d+1}(β_d/d^d) c^d symbolically and d-free for all d ≥ 2, with only β_d appearing on the edge; it is now cited at Conjecture 3.3.A★ and op:xi0-degree-d, and that problem is restated — no per-degree computation remains, and the residual is analytic rather than algebraic. (4) Every cross-reference to a proposition, conjecture, remark, definition, example or open problem rendered as "Theorem N" in v1.0–v1.5 (110 sites), because