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 degree-2 families), the three channels are non-cofinal: L(t) and BoT both fail where CC succeeds on the V_quad → P_III(D₆) detection task. 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preserved across versions); v1.1 (10.5281/zenodo.19941679) and v1.2 (10.5281/zenodo.19951331) are superseded by v1.3.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). V_quad's known P_III(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 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 B1/B2/B3 tiering. - New open problem op:cc-median-resurgence-execute (first entry of §9). Run median Écalle resurgence on the V_quad formal coefficient series at 5000 coefficients / dps 250, with ζ* = 4/√3 pinned at 200 digits. Resolves the residual Laplace step of op:cc-formal-borel for V_quad and would conditionally close the V_quad → P_III(D6) correspondence at the Stokes-data level under a documented normalisation map. 1–3 sessions of compute. - Refreshed open-problems list. op:cc-formal-borel — Status (v1.3): PARTIALLY DIAGNOSED by Theory-Fleet H4. op:xi0-d3-direct — Status (v1.3): Carried forward unchanged; tractability re-checked at 1–2 hours of computation, no halt. op:bridge-witness-tier — appended (v1.3) note that tier discrimination is now also informed by the umbrella's ‖Δ‖_Pet axis. op:cc-channel-existence (legacy) — appended (v1.3) note that the cubic case is now bound to PCF-2 v1.3 (Conjecture B5/B6 in their d = 3 restricted form). - Refreshed §10 SIARC sessions list with the 2026-05-{01,02} cohort: SIARC umbrella v2.0, PCF-2 v1.3, PCF2-SESSION-T2, Theory-Fleet H4, Theory-Fleet AGGREGATOR, PCF2-V13-RELEASE, SIARC-UMBRELLA-V2-RELEASE. - Bibliography expanded with four new entries: siarc_umbrella_v2 (concept 19885549, version 19965041), siarc_pcf2_v13 (concept 19936297, version 19963298), siarc_pcf2_session_t2 (bridge), siarc_theory_fleet_h4 (bridge; explicitly marked as theoretical prediction). - AI Disclosure refreshed to credit umbrella v2.0 / PCF-2 v1.3 / Theory-Fleet H4 absorption and to extend the AEAL-record reference to sessions/2026-05-{01,02}/. AEAL schema (v1.3 canonical form). Each per-session claims.jsonl entry conforms to the schema {"claim_id": ..., "task_id": ..., "claim": ..., "evidence_type": "file_inspection|computation|build", "dps": ..., "reproducible": true, "script": ..., "output_hash": " "}, with claim_id 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/. EBR ξ₀ upgrade and surface-type correction (added 2026-06-06) (1) ξ₀ identity upgrade. The ξ₀ = 2/√β₂ identity (Proposition 3.3.A) and its general-d form ξ₀ = d/β_d^(1/d) are, for positivity-hypothesis families b(n) > 0 (n ≥ 1), now proved for the physical generating-function object by the holonomic/positivity argument of the companion EBR preprint 10.5281/zenodo.20564080, §3 (D-finite G; leading coefficient a_{2d}(s) = d^d s^{2d}(d^d − β_d s), roots {0, R}; Pringsheim). This supersedes, for the positive-b case, the Conjecture-3.3.A* status of the cross-degree identity. Grade and scope as in the