Dimensional Reduction Mapping of Fermat's Hyper-Equations: A Complete Discrete Isolation Proof Based on the Logical Over-Premise Negation Theorem

This paper establishes a novel, self-contained programmatic framework for verifying Fermat's Last Theorem (FLT) by synergizing game-theoretic state-space isolation with modern predicate logic. Deviating from the heavily convoluted Modularity Theorem and arithmetic algebraic geometry frameworks dominant in traditional institutional academia, we implement a dimensional reduction strategy rooted in spatial conservation, formally projecting the classical Diophantine equation X^n + Y^n = Z^n into a lower-dimensional holomorphic discrete lattice. For the base boundary where n = 3, we deconstruct the non-continuous geometry of the cubic shell equations derived from spatial cube-stripping. We demonstrate that structural corner-blocks introduce an immutable, discrete microscopic residue—the "residual tail deficit." By enforcing a modular confinement network within the 3-adic local field Q_3 and establishing an exact inequality chain governing microscopic growth-rate disparities, the non-existence of non-trivial integer solutions for the cubic state space is strictly finalized. To systematically collapse hyper-dimensional exponents where n > 3, we deploy a formal operator embedding expressed as (X^(n/3))^3 + (Y^(n/3))^3 = (Z^(n/3))^3, mapping the hyper-equations back into the established n = 3 continuous geometric deficit domain. It is rigorously proven that any prospective non-integer extension under fractional domains inevitably violates the topological discrete isolation of the 3-adic field. This unified framework demonstrates that Fermat's original intuition of an "elegant and marvelous proof" can indeed be articulated through the synthesis of low-dimensional geometric invariants and localized algebraic valuations, offering a fresh, interdisciplinary paradigm for classical diophantine analysis. Research Note (June 2026) This publication represents the first public stage of the Fermat Exploration Project. The ideas developed in this paper were later expanded, reassessed, and critically reviewed in two subsequent publications: • Coherent Structural Universe and Defect-Conservation Ontology:A Logical Encirclement of Geometric Legitimacy and Higher-Power Inheritancehttps://doi.org/10.5281/zenodo.20608166 • The Search for Mirabilem:How an Independent Learner Explored Fermat's Last Theorem with AIhttps://doi.org/10.5281/zenodo.20840419 Readers interested in the complete development of the project are encouraged to consult all three publications as parts of a single exploration history.