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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.09028 |
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| _version_ | 1866915786968268800 |
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| author | Tochigi, Kanta |
| author_facet | Tochigi, Kanta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09028 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-existence of Information-Geometric Fermat Structures: Violation of Dual Lattice Consistency in Statistical Manifolds with $L^n$ Structure Tochigi, Kanta Information Theory 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. |
| title | Non-existence of Information-Geometric Fermat Structures: Violation of Dual Lattice Consistency in Statistical Manifolds with $L^n$ Structure |
| topic | Information Theory |
| url | https://arxiv.org/abs/2602.09028 |