Non-existence of Information-Geometric Fermat Structures: Violation of Dual Lattice Consistency in Statistical Manifolds with $L^n$ Structure

TL;DR

This paper demonstrates that certain information-geometric structures, modeled by $L^n$ constraints, cannot simultaneously satisfy dual lattice consistency, revealing fundamental incompatibilities in statistical manifolds with $L^n$ structures.

Contribution

It proves the non-existence of information-geometric Fermat structures for $n \,\geq\, 3$, highlighting a geometric obstruction due to Fourier transform incompatibilities.

Findings

Proves non-existence of dual lattice consistent structures for $n \ge 3$

Shows Fourier transform alters $L^n$ to $L^q$ spaces, breaking dual lattice symmetry

Identifies a fundamental incompatibility between local quadratic and global $L^n$ structures

Abstract

This paper reformulates Fermat's Last Theorem as an embedding problem of information-geometric structures. We reinterpret the Fermat equation as an $n$-th moment constraint, constructing a statistical manifold $\mathcal{M}_n$ of generalized normal distributions via the Maximum Entropy Principle. By Chentsov's Theorem, the natural metric is the Fisher information metric ($L^2$); however, the global structure is governed by the $L^n$ moment constraint. This reveals a discrepancy between the local quadratic metric and the global $L^n$ structure. We axiomatically define an "Information-Geometric Fermat Solution," postulating that the lattice structure must maintain "dual lattice consistency" under the Legendre transform. We prove the non-existence of such structures for $n \ge 3$. Through the Poisson Summation Formula and Hausdorff-Young Inequality, we demonstrate that the Fourier transform induces an alteration of the function family ($L^n \to L^q$, where $1/n + 1/q = 1$), rendering dual lattice consistency analytically impossible. This identifies a geometric obstruction where integer and energy structures are incompatible within a dually flat space. We conclude by discussing the correspondence between this model and elliptic curves.