An Edge–Borel Radius Theorem for Positive Polynomial Continued Fractions

[2026-09-23] Scope correction — the “no logarithmic factor” clause. This record states the coefficient asymptotic g_n ∼ C R^{−n} n^{γ−1} with no logarithmic factor, and separately identifies the arithmetic subvariety γ ∈ ℤ as the locus where the singularity is a pole rather than a branch point. Those two statements meet on that subvariety, and the first is not claimed there: for γ ∈ ℤ a resonance logarithm can occur. The clause is therefore scoped to γ ∉ ℤ. Off that subvariety — which includes the reference family b = 3n² + n + 1, with γ = 11/6 — the statement stands exactly as published. This correction is inherited: it follows the erratum issued the same day on EBR-Ib (10.5281/zenodo.20566465), and agrees with EBR-III (10.5281/zenodo.20684732), which establishes that the local monodromy at R is a semisimple complex pseudo-reflection for γ ∉ ℤ. What is NOT affected. The location R = dᵈ/β_d, the exponent γ = (d+1)/2 + b_{d−1}/β_d, the holonomy and single-dominant-singularity result, the branch/pole split itself, and the Lean 4 / Mathlib machine-checked layer at the granularity stated in §8 are all unchanged. The grade statement below remains load-bearing and is unmodified. Cluster statement (read with this notice). A dependency analysis of the 61-record deposit corpus finds this record inside a group of 18 records that cite one another in both directions across successive versions. The deposit metadata cannot establish an ordering within that group, so it cannot be determined from the record which way influence ran between its members. The group contains all three records whose headline claims are withdrawn or under review: CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089) LMP — An explicit exponential-period representation of the V_quad connection coefficient (concept DOI 10.5281/zenodo.20719042) This record is not one of the three, but it cites the companion and sits inside the group. This notice therefore does not assert that the other members of that group are unaffected. Corrections to them are being issued separately; each cites the others. Corrections issued together. This notice is issued in one batch with the corrections to the records below. Each carries its own notice, which this one does not summarise. CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635; correction notice, this batch) T2B — Two arithmetic classes of degree-(2,1) Trans-stratum continued fractions: a Birkhoff–Trjitzinsky / Gauss-continued-fraction dichotomy (concept DOI 10.5281/zenodo.19783311; description correction, this batch) EBR-Ib — The EBR Amplitude as a Connection Coefficient: Characterization and a Rigidity Dividing-Line Conjecture (concept DOI 10.5281/zenodo.20566465; erratum, this batch) LMP — An explicit exponential-period representation of the V_quad connection coefficient (concept DOI 10.5281/zenodo.20719042; correction notice, this batch) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089; corrected in its version 1.3, already deposited) For a degree-d polynomial continued fraction with unit numerators and partial denominator b(n) = β_d nᵈ + b_{d−1}n^{d−1} + ⋯ + b₀ (β_d > 0) satisfying the positivity hypothesis b(n) > 0 for all n ≥ 1, we determine the dominant Borel singularity of the associated generating function exactly and uniformly in the degree. Writing ξ₀ = d/β_d^{1/d} and R = ξ₀ᵈ = dᵈ/β_d, we prove that the genuine Gevrey-1 Borel transform B[ŷ_d](ζ) = Σ Q_n ζ^{dn}/(dn)! has its nearest singularity to the origin at |ζ| = ξ₀, with the s-plane generating function G(s) = Σ Q_n sⁿ/(dn)! holonomic and singular on its circle of convergence only at s = +R. The dominant singularity is a regular singular point of the order-2d annihilating operator, with local form G(s) ∼ A(1 − s/R)^{−γ} and exponent γ = (d+1)/2 + b_{d−1}/β_d; the coefficient asymptotic is g_n ∼ C R^{−n} n^{γ−1} with no logarithmic factor [2026-09-23 scope correction: this holds for γ ∉ ℤ. On the subvariety γ ∈ ℤ a resonance logarithm can occur and the stated form is not claimed there. See the notice at the head of this description.]. The singularity is an algebraic branch point for generic b and a pole on the explicit arithmetic subvariety where γ ∈ ℤ. Both the location R and the exponent γ are closed forms uniform in d. The result holds for the positivity-hypothesis families; the amplitude A is a global connection coefficient that we do not evaluate and that is not required by any downstream use in the companion program. GRADE STATEMENT (load-bearing; must remain in the public description). The general-d location and exponent are established by closed-form symbolic arguments (a leading-symbol computation and an O(1/n) ratio expansion), each independently corroborated by exact and high-precision numerical verification through degree 6; the algebraic layer is additionally machine-checked in Lean 4 / Mathlib with axiom cone {propext, Classical.choice, Quot.sound} and no `sorry`, at the item-by-item granularity stated in §8 (positivity for all degrees; the leading-coefficient factorization at degrees 2–5; the exhaustive root-set classification at degree 2 and the no-negative-root content at degrees 2–3; the γ-arithmetic and branch/pole split at the instances d = 2, 3, 5). This is a by-hand symbolic argument with finite machine and numerical verification, not a machine-checked proof quantified over the symbolic degree; that distinction is maintained throughout. Lean machine-checked granularity (five-way, stated precisely): positivity Q_n>0 for ALL degrees (degree-independent); leading-coefficient factorization a_{2d}(s)=d^d s^{2d}(d^d-beta_d s) at degrees 2-5; the exhaustive root-set {0,R} classification at degree 2; the no-negative-root content of Corollary 4.2 at degrees 2-3; gamma-arithmetic with branch/pole split at instances d=2,3,5. The gamma-law itself is symbolic with numeric verification through degree 6, NOT a Lean general proof.