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 bundle curvature. (5) Complete Invariant Pair: the pair (Theta_H, D) recovers the period exponent via nu = (D + Theta_H)/(2 Theta_H), verified on four Kodaira types. (6) Conformity Classification of Degenerations: D distinguishes additive reduction (meromorphic pole with residue 2nu(2nu-1)) from multiplicative reduction (log-essential singularity). (7) Conformity Cubic: D satisfies the autonomous first integral (D')^2 = 2(D+2Q)(D+4Q)^2, a degenerate elliptic curve of Kodaira type I_1. (8) CFT Central Charge: the node of the conformity cubic encodes the stress tensor with c = 24T/D_node = 1, matching the free boson CFT. (9) Schwarzian Identity: D + S(tau) = 2u^2 - 2pu - p^2/2 - p'. This extends the author's previously published conformity gradient theorem (DOI: 10.5281/zenodo.19301065) from a vanishing criterion into a complete classification tool for singular fibers of elliptic fibrations.