Fisher manifold geometry of helical cylindrical lattices: entropic deficit, scalar curvature, BKT divergence, UV–IR bridge, mutual information area-law, and the curvature–information connection

Fisher manifold geometry of helical cylindrical lattices: entropic deficit, scalar curvature, BKT divergence, UV–IR bridge, mutual information area-law, and the curvature–information connection Pavol BelobradIndependent Researcher, Žilina, SlovakiaApril 2026 AbstractWe study the information geometry of helical cylindrical lattices with N protofilaments and helical offset h. The microtubule geometry (N=13, h=3) serves as a structural template motivating the lattice topology; we do not model microtubule biophysics. Starting from the exact Bessel expansion of the partition function per node, we derive the entropic deficit C as a local geometric invariant independent of lattice size N, and compute the Fisher–Rao metric and its scalar curvature R on the (J₁, J₂, J₃) parameter space. All metric quantities are expressed as closed-form Bessel sums with no perturbative truncation. Main results: (1) R = +2.6245 at the reference point (β=0.5, J₁=J₂=1, J₃=0.8, h=3), stable to six significant digits; (2) R grows rapidly near the BKT transition with effective scaling R ~ β^4.79 (first computation of Fisher scalar curvature at a BKT transition); (3) helical topology inverts the curvature sign (R > 0 for helical vs. R < 0 for square lattices); (4) UV–IR Fisher manifolds share curvature sign only for N=13; (5) commensurability resonances governed by gcd(N,h). New results in version 3: (6) mutual information I(A:B) exhibits exact area-law scaling I = N · f(β, J₁, J₃, h) (analytical proof via Bloch decomposition); helical coupling increases boundary information density by 53% relative to the square lattice; (7) linear relationship R ≈ 5.9 · (I/N) − 0.088 (Pearson correlation +0.99) connecting Fisher scalar curvature to boundary mutual information.