Channel Theory for Polynomial Continued Fractions: Asymptotic Channels, the ξ₀ = 2/√β₂ Identity, and a Bridge Conjecture

Abstract

Correction notice (v1.5). This version demotes a claim. Clause (iii) of Theorem 3.7 -- the match between the CC-channel connection-coefficient ratio and the Riemann--Hilbert datum of P_V/D_5^(1) at the V_quad tau parameter -- was stated as part of a theorem through v1.4. It is not a theorem of this paper and is recorded at v1.5 as the open problem op:cc-stokes-closed-form (Section 9). The v1.4 description's statement that "both corrections are additive/clarifying; no channel-theoretic conjecture or numerical claim is retracted" does not describe the present version and is superseded by this notice. Reason. The surface-type premise this paper imported has itself been demoted: the V_quad companion paper records the D_5^(1) classification as Conjecture 4.1, not as an established reduction (10.5281/zenodo.20455089, read at v1.2). The Sakai Stratification (p. 4, Observation "two objects, two surfaces") further records that the object on which this paper computes xi_0 and rho and the object the D_5^(1) conjecture concerns are distinct, so a CC-channel datum cannot by itself settle the label. Withdrawn as non-evidential. The v1.4 sentence reporting that the Domb--Sykes sample radii drift "from 0.426 to 0.426 in the leading digits" is withdrawn with the paragraph containing it. The two endpoints agree to 2 decimal digits; rounding them to three creates the appearance of a stated range where the quoted numerals exhibit none. Unchanged. The xi_0 = 2/sqrt(beta_2) identity and the exact values xi_0 = 2/sqrt(3), rho = -11/6; the L(t)/BoT non-cofinality results; the channel definitions; and the EBR xi_0 addendum. No numerical value in this paper changes. Also unchanged from v1.4. The §7.4 Class B assignment remains recorded as op:classB-surface-determination (open), as in v1.4. Citations corrected. Five sentences citing the V_quad companion paper carried a version pin (10.5281/zenodo.20455090, v1.0) together with the label "Conjecture 4.1". That label does not occur in the pinned version, where the object numbered 4.1 is stated as a theorem; it was demoted to a conjecture at v1.2. The citations now carry the concept DOI with the version read (10.5281/zenodo.20455089, read at v1.2). No claim in this paper depended on the version cited. Second ground. The computational run cited for clause (iii) [CC-PIPELINE-G, 2026-05-01] stores a recorded comparison between its computed Borel radius and xi_0 that its own session verdict did not incorporate. The two quantities are expressed in the same variable, and the recorded factor of approximately two is accounted for by an explicit halving in the source. A residual discrepancy of about 1.7% remains unexplained, and the stored coefficients are insufficient to re-derive the compared quantity. Clause (iii) is therefore moved to open problems on this ground as well as on the ground given above. Channel Theory for Polynomial Continued Fractions: Asymptotic Channels, the ξ₀ = 2/√β₂ Identity, and a Bridge Conjecture (v1.5). Position. We propose, define, and catalogue *asymptotic channels* for sequences arising from polynomial continued fractions (PCFs), formalised as triples (D, T, S) consisting of a formal-series space D, an asymptotic gauge T, and an analytic-continuation section S. Three concrete channels appear in the SIARC stack: the recurrence-parameter channel L(t), the Borel-of-trans-series channel BoT, and the connection-coefficient channel CC. Empirically (six Δ < 0 degree-2 families), the three channels are non-cofinal: L(t) and BoT both fail where CC succeeds on the V_quad → Painlevé V / Sakai D₅^(1) detection task (that surface-type assignment is conjectural; companion paper, Conjecture 4.1). This outline is a top-level posting (concept DOI 10.5281/zenodo.19941678, preserved across versions); v1.1 (10.5281/zenodo.19941679) and v1.2 (10.5281/zenodo.19951331) are superseded by v1.3. Universality findings (CC-PIPELINE-G + PCF2-SESSION-Q1, 2026-05-01). For every degree-2 PCF (1, b) in scope with b(n) = β₂ n² + β₁ n + β₀ (β₂ > 0), a Newton-polygon analysis of the order-2 ODE governing the partial-denominator generating function f(z) = Σ Q_n zⁿ at z = 0 yields a single slope-1/2 edge of multiplicity 2 with characteristic polynomial 1 − (β₂/4) c² and hence Borel-singularity radius ξ₀ = 2/√β₂; the secondary indicial exponent at the irregular singular point in the uniformiser u = √z is ρ = −3/2 − β₁/β₂ (Proposition 3.3.A). At d = 4, the same Newton-polygon construction gives a single slope-1/4 edge with characteristic polynomial 1 − (β₄/4⁴) c⁴ and hence ξ₀ = 4/β₄^(1/4), verified at dps = 80 across 8 quartic representatives with spread 0 (Proposition 3.3.A', PCF2-SESSION-Q1). This promotes the cross-degree formula to a universality conjecture (Conjecture 3.3.A*): ξ₀(b) = d/β_d^(1/d) for every degree-d PCF in scope, proven at d = 2, verified at d = 4, with d = 3 verification deferred to op:xi0-d3-direct. The earlier v1.1 candidate c(d) = 2√((d−1)!) — which agrees with d = 2 by coincidence — is empirically falsified at d = 4 (predicts ≈ 4.899, measured 4, ≈22% disagreement; Remark 3.3.E). The map (α₁, α₀, β₂, β₁, β₀) ↦ ξ₀ is non-injective: QL01 and QL02 share both ξ₀ = 2 and ρ = −5/2 and split only at a₂ (Proposition 3.3.B). The K-SCAN marginal coincidence between V_quad and QL15 (both β₂ = 3, both ξ₀ = 2/√3 to 200 digits) is structurally explained by Proposition 3.3.A — same β₂ forces same ξ₀ without forcing same Painlevé label (Remark 3.3.C). ξ₀ and ρ are recovered through the CC channel in exact closed form, and the resulting algebraic identity was checked to 200 digits; neither invariant discriminates surface type, the Stokes-level identification remains open, and V_quad's Painlevé V / Sakai D₅^(1) classification is conjectural (companion paper, Conjecture 4.1; 10.5281/zenodo.20455089, read at v1.2), modulo Borel-Laplace summation of the formal coefficient series (Theorem 3.3.D). Bridge conjecture and open problems. The bridge identity Φ_b connecting the CC datum (ξ₀, ρ, a₁, a₂, …) to a Painlevé P-class label is functorial in the coefficient tuple rather than in the discriminant alone, and stratifies into three observed tiers: B1 (β₂-only witness; the weakest tier, witnessed by the universal ξ₀ identity), B2 (β₂ plus one secondary; the QL01/QL02 splitting case), B3 (full coefficient tuple; the general case). The K-SCAN reject of QL15 already pushes V_quad's tier-B1 witness down to B2 or higher. We list nine open problems, including a refined op:cc-formal-borel (numerical Borel-Laplace summation of the V_quad formal coefficient series at convergent tail accuracy, the residual after the algebraic-identity step), op:xi0-degree-d narrowed to its remaining direct-d=3 component (op:xi0-d3-direct, a 1–2 hour Newton-polygon test on one cubic representative per Galois bin), and the cross-referenced PCF-2 op:b4-degree-d narrowed to d ≥ 5 (PCF2-SESSION-Q1 confirmed A = 2d = 8 at d = 4, 60/60). Tier-discrimination, no-go proof for L(t) at Δ < 0 degree-2, monodromy / Riemann–Hilbert closure, channel moduli structure, and the BOREL CHANNEL ANOMALOUS flag complete the program. This outline supplements PCF-1 v1.3 (10.5281/zenodo.19937196), PCF-2 v1.1 (10.5281/zenodo.19939463), PCF2-SESSION-Q1 (SIARC bridge sessions/2026-05-01/PCF2-SESSION-Q1/), and the SIARC umbrella master conjecture (10.5281/zenodo.19885550). Empirical record on the SIARC bridge: github.com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-01/ (folders BOREL-CHANNEL-K-SCAN, CC-PIPELINE-F, CC-PIPELINE-G, PCF2-SESSION-Q1, CHANNEL-THEORY-V11, CHANNEL-THEORY-V12). Method. AEAL/SIARC multi-agent methodology (cf. Paper P8). AI tools (GitHub Copilot powered by Anthropic Claude Opus 4.7, and Anthropic Claude) used for literature synthesis, prose drafting, computation orchestration, and consistency-checking; all conjectures, theorem statements, and editorial decisions are the author's. All numerical claims trace to exact computations stored under sessions/2026-05-01/ on the SIARC bridge with AEAL claim entries in claims.jsonl. v1.3 update (2026-05-02). v1.3 is a purely additive framing revision: no v1.0 / v1.1 / v1.2 theorem, proposition, or conjecture is rewritten or retracted. Three external inputs are absorbed. What is new in v1.3 (release-level summary): - New §3.5 "Median Écalle resurgence as the residual Laplace step (v1.3)". One numbered remark (rem:h4-median) records the Theory-Fleet H4 prediction (sessions/2026-05-01/THEORY-FLEET/H4/) as a *theoretical prediction at HIGH confidence* — explicitly NOT an executed measurement — that median Écalle resurgence on the V_quad formal coefficient series at 5000 coefficients with mpmath dps = 250 will extract the leading alien amplitude S_{ζ*} at ζ* = 4/√3 to 30–50 digits, with a central forecast of ~40 digits. The recipe (fit branch exponent from large-order tail; extract Stokes constant via half-Stokes prescription; cross-check by local Borel singular-germ fitting) and the cross-reference to the new op:cc-median-resurgence-execute are recorded; H4 explicitly does not forecast a closed-form S_{ζ*}. - New §3.6 "The cross-degree invariant triple of the SIARC umbrella (v1.3)". The post-T2 refactor of the SIARC umbrella v2.0 (10.5281/zenodo.19965041, [SIARC v2.0, §4.4]) packages the cross-degree SIARC stack as a bridge functor Φ : PCF(1,b) ↦ (Δ_d(b), ‖Δ‖_Pet(τ_b), ξ_0(b)). The Borel-radius axis carried by Channel Theory since v1.0 is the third coordinate. The modular-discriminant axis is established empirically by PCF-2 v1.3 / PCF2-SESSION-T2 at Spearman ρ = +0.638, p_Bonferroni = 8.6×10⁻⁶ on n = 50 cubic families, beating the bare log|j| baseline by ~30× in Bonferroni p. Channel Theory adds nothing structurally new on the Δ_d or ‖Δ‖_Pet axes (those are PCF-1/PCF-2/umbrella-v2.0 territory); §3.6 records the compatibility expectation that any successful bridge identity Φ_b will factor through Φ on the relevant axes, and a (β_d, ‖Δ‖_Pet, ξ_0)-coordinatised refinement of the B