Fisher manifold geometry of helical cylindrical lattices: entropic deficit, scalar curvature, BKT divergence, and the UV–IR bridge

We study the Fisher–Rao information geometry of a 2D classical XY model on a helical cylindrical latticewith N protofilaments and helical offset h. The microtubule geometry (N = 13, h = 3) is used only as astructural template motivating the lattice topology; no biophysical claim is made. We express the partitionfunction per node as an exact Bessel sum (eq. 1) and the Fisher–Rao metric on the coupling parameterspace (J₁, J₂, J₃) as a closed-form Bessel expression (eq. 3). These two results are rigorously verified byindependent recomputation, with the m=0 term contributing 99.42% of the Bessel sum at the referencepoint and the analytical metric matching the numerical Hessian of ln z to single-precision roundoff.