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 thirteen results connecting information geometry, Hodge theory, and conformal field theory through a single computable scalar invariant D (t) = W''/W. (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) is algebraically determined by the Picard-Fuchs equation via D = 2u² - 2pu - 2q. (4) Conformity-Hodge Decomposition: D = 4u² - ThetaH where ThetaH is the Hodge bundle curvature. (5) Complete Invariant Pair: the pair (ThetaH, D) recovers the period exponent via nu = (D + ThetaH) / (2 ThetaH), 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: for constant Q, D satisfies (D') ² = 2 (D+2Q) (D+4Q) ², a degenerate elliptic curve of Kodaira type I₁ with discriminant Delta = 0. (8) CFT Central Charge: the node of the conformity cubic encodes the stress tensor with c = 24T/Dₙode = 1, matching the free boson CFT. (9) Schwarzian Identity: D + S (tau) = 2u² - 2pu - p²/2 - p'. Version 2 adds three new results proved since v1, all verified symbolically (SymPy: LHS - RHS = 0): (10) Covariant First Integral: (D' + 2Q') ² = 2 (D+2Q) (D+4Q) ², valid for ALL variable Q (t), extending the conformity cubic from constant Q to universal. This shows type I₁ and c = 1 are universal for all elliptic fibrations. The variable-Q correction is a gauge shift D' -> D' + 2Q', not a curve deformation. (11) Conformity Connection: A = (ThetaH' - D') /2 dt = 2Q' dt, the natural gauge structure on the conformity gradient. The connection vanishes iff Q is constant iff D and ThetaH evolve at the same rate. (12) Geometric First Integral: (D' + ThetaH') ² = 8 (D+2Q) (D+4Q) ², a purely geometric identity closing the system without reference to the period. The variable-Q conjecture from v1 is replaced: the conformity cubic does not deform but gauge-shifts. This extends the author's conformity gradient theorem (DOI: 10. 5281/zenodo. 19301065). Arithmetic data generated using the cubic congruence formula (DOI: 10. 5281/zenodo. 19439021).