For a degree-d polynomial continued fraction with unit numerators and partial denominator b (n) = βd nᵈ + b₃−₁n^d−1 + ⋯ + b₀ (βd > 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/βd^1/d and R = ξ₀ᵈ = dᵈ/βd, we prove that the genuine Gevrey-1 Borel transform Bŷd (ζ) = Σ Qₙ ζ^dn/ (dn) ! has its nearest singularity to the origin at |ζ| = ξ₀, with the s-plane generating function G (s) = Σ Qₙ 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 + b₃−₁/βd; the coefficient asymptotic is gₙ ∼ 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 Qₙ>0 for ALL degrees (degree-independent) ; leading-coefficient factorization a₂₃ (s) =dᵈ s^2d (dᵈ-betad 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.
Papanokechi (Sat,) studied this question.