An Edge–Borel Radius Theorem for Positive Polynomial Continued Fractions

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. 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.