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.

翻译：本文将费马大定理重新表述为信息几何结构的嵌入问题。我们将费马方程重新解释为$n$阶矩约束，通过最大熵原理构造了广义正态分布统计流形$\mathcal{M}_n$。根据Chentsov定理，其自然度量为Fisher信息度量（$L^2$）；然而全局结构受$n$阶矩约束支配，揭示了局部二次度量与全局$L^n$结构之间的差异。我们公理化地定义了“信息几何费马解”，要求其格结构在Legendre变换下必须保持“偶格子一致性”。我们证明了当$n \ge 3$时此类结构不存在。通过Poisson求和公式与Hausdorff-Young不等式，我们证明了Fourier变换会导致函数族从$L^n$变为$L^q$（其中$1/n + 1/q = 1$），使得偶格子一致性在分析上不可实现。这揭示了在双平坦空间中整数结构与能量结构不相容的几何障碍。最后我们讨论了该模型与椭圆曲线之间的对应关系。