Chern--Simons Fluctuations and Information Geometry in Discrete Electromagnetism

Abstract

We construct helicity-conditioned statistical states for a Whitney-discretized electromagnetic field on a closed oriented three-manifold. The simplicial de Rham complex provides exact discrete gauge symmetry, while the Whitney inner product separates exact, harmonic, and coexact sectors. The spatial Abelian Chern--Simons functional is gauge invariant and depends only on the coexact potential. After fixing harmonic modes, we introduce a helicity-biased Gaussian ensemble on the reduced electromagnetic phase space and derive explicit formulas for its admissible parameters, partition function, mean helicity, relative entropy, and Fisher information. The distribution uniquely minimizes relative entropy under a prescribed mean-helicity constraint. Its helicity susceptibility equals the variance of the discrete Chern--Simons functional and controls the local distinguishability of neighboring statistical states. Because magnetic helicity is generally not conserved under unconstrained Maxwell dynamics, these states represent conditioned inference rather than dynamical equilibrium.