The Conformity Gradient on Calabi--Yau Moduli: From Statistical Rigidity to an Integrable System on the Fermat Quintic

We prove that five geometric objects — the conformity gradient D, Hodge curvature ThetaH, Picard-Fuchs potential Q, Amari tensor A, and Schwarzian S — form a complete integrable system on the complex-structure moduli space of Calabi-Yau manifolds. Starting from two rigidity theorems for the Sasaki-Dombrowski lift under Amari alpha-connections, we establish D (t) = W''/W as a complete information-geometric invariant for the Kodaira classification. We prove the covariant first integral (D'+2Q') ² = 2 (D+2Q) (D+4Q) ², valid for all variable Q (t), showing that the conformity cubic is universally of type I₁ with central charge c = 1. The conformity connection A = (ThetaH' - D') /2 dt provides a natural gauge structure. We then prove the Amari-Hodge identity A = -ThetaH' and the Amari-Schwarzian cubic (A+S') ² = 2 (D+S) (D+2S) ², establishing that the Amari-Chentsov tensor is completely determined by the conformity dynamics. Version 3 adds three new theorems: (14) Amari-Hodge identity A = -ThetaH', (15) Amari-Conformity relation A + D' + 4Q' = 0, and (16) Amari-Schwarzian cubic. All identities verified symbolically (SymPy: LHS - RHS = 0) and numerically on the Fermat quintic threefold using the full four-period symplectic partition function Z = Im (iPi†SigmaPi), with residuals at 10^-40. Pattern 3 (higher-order mirror-asymmetry invariant) is proved: the Amari tensor on CY3 moduli satisfies the conformity cubic. This extends the author's conformity gradient theorem (DOI: 10. 5281/zenodo. 19301065) and cubic congruence formula (DOI: 10. 5281/zenodo. 19439021).