Removing the Positivity Hypothesis: Edge\u2013Borel 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) = \u03a3 Q_n s\u207f/(dn)! of a degree-d polynomial continued fraction, under the positivity hypothesis b(n) > 0 for all n \u2265 1, which entered the proof through Pringsheim's theorem. Here we remove that hypothesis. We show that, for any b with positive leading coefficient \u03b2_d > 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 = d\u1d48/\u03b2_d, of regular-singular (Fuchsian) type with local exponent \u2212\u03b3, \u03b3 = (d+1)/2 + b_{d\u22121}/\u03b2_d. Two facts make this possible. First, the holonomic annihilator's leading coefficient a_{2d}(s) = d\u1d48 s^{2d}(d\u1d48 \u2212 \u03b2_d s) depends only on \u03b2_d, 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 Q_n, which we establish by identifying Q_n 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 \u03b2_d > 0 has an entire Borel transform. The positivity hypothesis is therefore removable.\n\n**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 \u2014 equivalently the absence of an entire Borel transform \u2014 is proved d-uniformly, **modulo classical continued-fraction and recurrence theory** (Pincherle's theorem, the Perron\u2013Kreuser dominant/minimal dichotomy, Seidel\u2013Stern 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 56\u201367 digits, including nine odd-degree families. The local exponent \u2212\u03b3 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.