Structural proof and numerical verification of the exact two-sided growth law for the convergent denominators q_n of the quadratic polynomial continued fraction V(A,B,C) = 1 + K_{n>=1} 1/(A n^2 + B n + C), for integers A,B,C >= 1. With b_k = A k^2 + B k + C and the standard continuant q_0=1, q_1=b_1, q_n = b_n q_{n-1} + q_{n-2}, we prove log q_n = n log A + 2 log(n!) + (B/A) log n + (K_Gamma + delta) + kappa/n + O(1/n^2), where K_Gamma = -log(Gamma(1-r1) Gamma(1-r2)) is the elementary naive-product constant (r1,r2 the roots of A x^2 + B x + C; at B=0 it equals log(sinh(pi sqrt(C/A)) / (pi sqrt(C/A)))), delta = log R_inf > 0 with R_inf = lim q_n / prod_{k<=n} b_k is a strictly positive continued-fraction correction from the +q_{n-2} term, and kappa = (1/2)(B^2/A^2 + B/A - 2C/A). This sharpens the deposited one-sided denominator-growth bound log Q_n >= n log n - O(n) (which suffices for Euler's irrationality criterion and underlies the deposited 482-constant catalogue) to an exact expansion through O(1/n) with an explicit additive constant. No closed form is claimed for delta (an 80-digit PSLQ search for the triple (1,0,1) returns none). Epistemic status (SIARC four-class): STRUCTURAL (complete elementary hand proof, not yet machine-checked in Lean), modulo one standard cited asymptotic (DLMF 5.11.13 Gamma-ratio expansion); VERIFIED numerically (mpmath, 50-60 digits) on seven triples, with the B=0 cross-check K_Gamma = sinh form holding to 50 digits. Changes in v1.1 (file-only): Replaced the manuscript PDF with the typeset repository build. The reference to the companion "Logarithmic Ladder" paper now carries its Zenodo concept DOI (10.5281/zenodo.19491767) in place of an internal placeholder note; an ORCID and code/data-availability + AI-assistance statements were added. No change to the mathematical content, results, or epistemic grading; the concept DOI (10.5281/zenodo.20564681) is unchanged.