The Conformity Gradient: A Computable Invariant Measuring Information-Geometric Deviation on Elliptic Curve Moduli
Nicholas Daniel Maino · Zenodo (CERN European Organization for Nuclear Research) · 2026
We define a new scalar invariant on the moduli space of one-parameter families of elliptic curves, measuring the deviation between the score-squared Fisher-Rao metric and the Weil-Petersson metric. For a family with real period w(t), the conformity gradient is D(t) = (w^2)''(t) / w(t)^2. We prove D(t) = 0 for all t if and only if w(t)^2 is affine in t, which is generically false for non-isotrivial families. Numerical verification on y^2 = x^3 + tx + 1 confirms D is nonzero with |D|/|Y| ~ 95, where Y is the Yukawa coupling. Applications to cryptographic curve health verification, numerical period stability, and natural gradient correction on moduli spaces are discussed. Integration with a multi-engine field-native algebraic verification pipeline is described.