Removing the Positivity Hypothesis: Edge2013Borel Location and Type for Sign-Varying Polynomial Continued Fractions

EBR-I established the location and type of the dominant Borel singularity of the generating function G (s) = 03a3 Qₙ s207f/ (dn) ! of a degree-d polynomial continued fraction, under the positivity hypothesis b (n) > 0 for all n 2265 1, which entered the proof through Pringsheim's theorem. Here we remove that hypothesis. We show that, for any b with positive leading coefficient 03b2d > 0 and lower coefficients of arbitrary sign, the conclusions of EBR-I hold verbatim: G is holonomic with a single dominant singularity at s = R = d1d48/03b2d, of regular-singular (Fuchsian) type with local exponent 221203b3, 03b3 = (d+1) /2 + b₃₂₂₁₂₁/03b2d. Two facts make this possible. First, the holonomic annihilator's leading coefficient a₂₃ (s) = d1d48 s^2d (d1d48 2212 03b2d s) depends only on 03b2d, so the localization of singularities to 0, R is independent of the lower coefficients' signs. Second, although Pringsheim no longer applies, the requirement that G be genuinely singular at R (equivalently, that the connection coefficient C be nonzero) reduces to a growth lower bound on Qₙ, which we establish by identifying Qₙ as the dominant solution of the Wallis recurrence via classical continued-fraction theory. The entire/singular dichotomy is resolved structurally: the minimal solution of the recurrence yields an entire Borel transform (C = 0), but the canonical continued-fraction denominators are never the minimal solution, so no polynomial-continued-fraction family with 03b2d > 0 has an entire Borel transform. The positivity hypothesis is therefore removable. **Grade statement (read before the results). ** The localization of singularities to 0, R is proved symbolically and is independent of lower-coefficient signs (verified across sign-varying families). The nonvanishing of the connection coefficient C 2014 equivalently the absence of an entire Borel transform 2014 is proved d-uniformly, **modulo classical continued-fraction and recurrence theory** (Pincherle's theorem, the Perron2013Kreuser dominant/minimal dichotomy, Seidel2013Stern convergence): these standard results are invoked, not re-derived or machine-checked. A constructive numerical witness (Miller backward recurrence exhibiting the entire minimal solution alongside the dominant physical solution) corroborates the proof across eleven sign-varying families to 56201367 digits, including nine odd-degree families. The local exponent 221203b3 and radius R inherit their grades from EBR-I (symbolic, verified through degree 6). The amplitude C itself is not evaluated (only its nonvanishing is established) ; its value is the open problem of EBR-II.