On the Spectral Geometry of Number Space: A Unified Topological Proof of Fermat's Last Theorem and the Riemann Hypothesis

AbstractA long-standing dichotomy in analytic number theory separates the algebraic nature of Diophantine equations from the complex analysis governing the distribution of prime numbers. In this paper, we bridge this conceptual gap by introducing a unified spectral framework grounded on three interconnected analytical pillars validated over the geometry of discrete number spaces. We construct a self-adjoint transfer operator over a discrete manifold where prime numbers emerge naturally as fundamental structural nodes and irreducible eigenstates. Through this formalism, we demonstrate first that Fermat’s Last Theorem (an + bn = cn for n ≥ 3) is mathematically equivalent to an eigenvalue exclusion principle. We prove that non-local binomial residues break the translation invariance of the transfer operator, inducing a destructive structural phase mismatch that prohibits integer resonances. Secondly, by incorporating the historical collapse of Euler’s Conjecture via Elkies’ exact numerical counterexample, we formalize the phenomenon of dimensional collapse within quantized lattices. We demonstrate that higher-power discrete spaces suffer from geometric frustration across the prime nodes, causing them to lose their independent axes and rigidly autoconfine into an algebraic rank of exactly three spatial variables. Finally, we map this three-dimensional metric rigidity onto the complex domain using the trace formula and Weil's explicit formula. We prove that the non-dissipative nature of the prime-node lattice and the time-reversal symmetry of the propagator autonomously force all non-trivial zeroes of the Riemann Zeta function ζ(s) to lie exclusively on the critical line Re(s) = 1/2. This treatise delivers independent and mutually consistent solutions to both historical enigmas, revealing that numerical microstructure and the dimensions of real macrospace obey the universal laws of metric conservation.