Non-existence of Information-Geometric Fermat Structures: Violation of Dual Lattice Consistency in Statistical Manifolds with $L^n$ Structure
This work provides a novel geometric perspective on a classical number theory problem, but the result is primarily theoretical and of interest to researchers in information geometry and mathematical physics.
The paper reformulates Fermat's Last Theorem as an embedding problem in information geometry, proving that for n ≥ 3, no statistical manifold with L^n moment constraint can maintain dual lattice consistency under the Legendre transform, thus showing a geometric obstruction analogous to Fermat's Last Theorem.
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.