Geometric Rejection of Diophantine Coexistence in Higher Dimensions: A Proof of Fermat's Last Theorem via the Grand Unified Dimension Equation

This paper provides a rigorous geometric and numerical proof showing that non-trivial integer solutions to Fermat’s LastTheorem, Xn + Y n = Z n, cannot exist for any exponent n ≥ 3. While two-dimensional space (n = 2) possesses a fluiddegree of freedom that permits concentric geometric scaling (gnomonic expansion), higher-dimensional modular grids arerigidly governed by an invariant isotropic structural synchronization. We establish that all spatial dimensions arebound by a singular overarching principle denoted as the Grand Unified Dimension Equation: a n + a n(a − 1) = a n+1. Inthe plane (n = 2), the structural noise term (a − 1) is perfectly cancelled out and neutralized by the even power of thebounding layer (Y 2 ), enabling the timeless harmony of the Pythagorean theorem. However, for any higher dimension n ≥3, this cancellation becomes structurally impossible due to increased spatial rigidity, forcing the total aggregatevolume (Z) to be pushed up into a strictly higher degree (n+ 1). Consequently, the simultaneous alignment of all threeactors as equivalent n-th degree hypercubes is structurally prohibited, completing the geometric proof of Fermat’s LastTheorem.