The Conformity Gradient on Elliptic Fibrations: From Statistical Rigidity to Kodaira Classification and CFT

We study the statistical geometry of elliptic fibrations equipped with the probability measure p proportional to |Omega|^2. We prove ten results connecting information geometry, Hodge theory, and conformal field theory through a single computable scalar invariant. (1) Universal Rigidity Lemma: the Sasaki-Dombrowski volume form is alpha-invariant for any Riemannian base. (2) Flat-Base Integrability: the lifted almost-complex structure is integrable for all alpha on flat statistical manifolds. (3) Conformity-Picard-Fuchs Correspondence: D(t) = W''/W is algebraically determined by the Picard-Fuchs equation via D = 2u^2 - 2pu - 2q. (4) Conformity-Hodge Decomposition: D = 4u^2 - Theta_H where Theta_H is the Hodge…