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

Abstract

Erratum notice (v1.6). This version repairs five defects in the v1.5 record. It is a build, labelling and open-problem erratum: no theorem, proposition, conjecture, or numerical value of v1.5 is altered by it. Concept DOI is preserved (10.5281/zenodo.19941678); v1.5 (10.5281/zenodo.22717792) is superseded. (1) Wrong build attached. The PDF attached to v1.5 was the draft build. It carries a "DRAFT -- NOT DEPOSITED" watermark on every page; its title-page date line had not advanced past "2026-06-07 (v1.4, draft)"; and its "What is new" changelog series ended at v1.3 while the record stood at v1.5. The body of that build was v1.5 content -- the v1.5 correction notice is present and clause (iii) is recorded as open -- so no reader was given superseded mathematics; the defect is in the front matter and the watermark, not in the text. The v1.6 build removes the watermark, advances the date line, and writes the missing "What is new in v1.4", "v1.5" and "v1.6" entries. This is the same defect class as 10.5281/zenodo.22739009 (wrong build attached), and is therefore repaired by a new version rather than by a metadata edit. (2) op:xi0-d3-direct is withdrawn as vacuous. Versions 1.2--1.5 carried an open problem proposing to select one cubic representative per Galois bin (C3, S3-real, CM), measure the Borel-singularity radius ξ₀ at the irregular singular point of L₃f = (1 − zB₃(θ+1) − z²)f, and compare with the prediction c(3) = 3. That test cannot fail, and cannot succeed, in the discriminating sense it was posed for: the observable does not depend on the structure it was meant to probe. Under a sweep of the lower-order coefficients β₂, β₁, β₀ over −1000..+999 with β₃ held fixed, ξ₀ moves by 6.85×10⁻¹⁸, while varying β₃ alone moves it by 1.26×10⁻¹. The Galois bin is a property of the splitting field of b; ξ₀ reads β₃ alone. The proposed measurement would therefore have returned agreement in all three bins whatever the bin structure was, and the agreement would have carried no information about it. The problem is withdrawn and replaced by op:xi0-singularity-type, which states the analytic question the withdrawn problem stood in front of -- the type of the Borel singularity, which is not determined by its radius. A remark (rem:d3-vacuous) records the withdrawal and the measurement behind it. Consequently superseded in the text of this description. The sentence in the "Universality findings" paragraph below reading "proven at d = 2, verified at d = 4, with d = 3 verification deferred to op:xi0-d3-direct", and the sentence in the "Bridge conjecture and open problems" paragraph describing "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)", both stand as written for v1.2--v1.5 and do not describe v1.6. The d = 3 deferral is withdrawn, and the cost estimate it carried was a cost for work that could not have discriminated what it was posed to discriminate. Those paragraphs are retained below unedited as the record of what v1.2--v1.5 stated. (3) Citation gap. 10.5281/zenodo.20542162 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 was deposited on 2026-06-04, three months before v1.5 restated the d = 3 case as outstanding work, and v1.1--v1.5 of this outline cite it zero times. It is now cited at Conjecture 3.3.A* and at op:xi0-degree-d, and op:xi0-degree-d is restated: no per-degree computation remains, and the residual is analytic rather than algebraic. The grade is recorded in the text as symbolically derived, which is explicitly weaker than the machine-checked PROVEN used elsewhere in this corpus; this addition is a citation repair, not a strengthening of any claim. It is distinct from, and does not duplicate, the EBR ξ₀ upgrade recorded further below (10.5281/zenodo.20564079), which concerns the positivity/holonomic argument for the physical generating-function object. (4) Cross-references named the wrong object class. The theorem-like environments in this manuscript share a single counter, and the cross-referencing package derives a reference's name from the counter rather than from the environment. Every cross-reference to a proposition, conjecture, remark, definition, example or open problem therefore rendered as "Theorem N". The numbered blocks themselves were always correct -- the document declares "Proposition 3.2" and "Conjecture 3.3.A*" properly -- but the references to them did not. In v1.5 as shipped this affected 110 sites: 88 rendered occurrences of "Theorem N.N" against 4 of "Proposition N.N", including 22 references to a proposition and 8 to the cross-degree universality conjecture. In the v1.6 build the counts are 22 "Theorem" (the genuine theorems), 38 "Proposition", 39 "Conjecture", 24 "Remark". No number, statement, or ordering changes; only the word naming the object class. This defect is of the same kind as the one the v1.5 correction notice exists to repair -- a claim being referred to as a theorem when it is not one -- and it is recorded here rather than corrected silently for that reason. Readers of v1.0--v1.5 should treat "Theorem N.N" in running text as naming whatever class the numbered block itself declares. (5) Falsifiers added to two open problems. op:cc-median-resurgence-execute and op:cc-formal-borel carried published cost estimates with no stated failure condition. Both now carry an explicit falsifier with a numeric threshold, stated as a claim about what the method can resolve rather than as a tolerance chosen after the fact. op:cc-median-resurgence-execute additionally carries a vacuity pre-check to be run before its compute budget is committed: hold β₂ fixed, vary the lower-order coefficients, and confirm that the alien amplitude S_{ζ*} actually moves. An observable that depends on less data than the hypothesis it is posed to test cannot test it -- this is the transferable content of defect (2), and it applies to open problems in this corpus that have not yet been examined for it. All surviving cost estimates in Section 9 are now explicitly marked unverified. What is NOT claimed by v1.6. No theorem, proposition, conjecture, or numerical value of v1.5 is retracted, weakened, or strengthened. The ξ₀ = 2/√β₂ identity and the exact values ξ₀ = 2/√3, ρ = −11/6 stand; the L(t)/BoT non-cofinality results stand; the channel definitions stand; the v1.5 demotion of Theorem 3.7(iii) stands unchanged. Defect (2) is a finding about a test, not about a claim: it does not assert that c(3) = 3 is false, and it does not assert that it is true. It asserts that the measurement published as the way to check it could not have distinguished the two. No claim in this paper depended on that measurement having been performed. Provenance of this erratum. The defects above were found by a study targeting this record's own pre-registered open-problems queue. Before the queue was read, the attached v1.5 PDF was verified by content to be a v1.5-content build rather than a stale earlier build, so that the quoted problem statements and cost estimates are known to come from the version the record claims. The v1.6 build was produced from the frozen v1.5 source; an unmodified control build of that source reproduces the deposited v1.5 PDF at the same byte length with all content markers matching, so the differences in the v1.6 build are attributable to the edits listed above and not to the build environment. The remainder of this description is the v1.5 text, retained unedited except where the v1.6 notice above states that a sentence has been superseded. 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 computat