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

Channel Theory for Polynomial Continued Fractions: Asymptotic Channels, the ξ₀ = 2/√β₂ Identity, and a Bridge Conjecture (v1. 2). 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), a Newton-polygon analysis of the order-2 ODE governing the partial-denominator generating function f (z) = Σ Qₙ 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/βd^ (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 Vquad 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). Vquad's known PIII (D₆) reduction is recovered through the CC channel as an exact algebraic identity to 200 digits, modulo Borel-Laplace summation of the formal coefficient series (Theorem 3. 3. D). Bridge conjecture and open problems. The bridge identity Φb 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 Vquad'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 Vquad 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 "}, with claimᵢd and verdict optional. Numerical claims with no hashable script (e. g. theoretical predictions cited from sister sessions) are admissible only if the source session's claims. jsonl entry carries the underlying hash. No conjecture or theorem from v1. 0–v1. 2 is retracted. v1. 3 is purely additive framing wrap. Concept DOI preserved (10. 5281/zenodo. 19941678) ; v1. 0, v1. 1, and v1. 2 are superseded by this version. The 17-page v1. 3 PDF integrates all v1. 3-narrative inline (§3. 5, §3. 6, §10 v1. 3 subsection, refreshed §9 open-problems entries, abstract sentence, AI disclosure paragraph) and is mirrored on the SIARC bridge at https: //github. com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-02/CHANNEL-THEORY-V13-RELEASE/.