Unified D10Z Resolution of the Legendre and Polignac Conjectures via Fractal Nodal Resonance

This paper presents a unified D10Z framework for resolving both Legendre's and Polignac's conjectures, using a quantum fractal mesh (TTAϵ) operating at scale GM = 10⁻⁵¹ m. Primes are modeled as coherent nodal attractors, and both conjectures emerge as necessary coherence conditions of the Spiderweb Fabric. Legendre’s conjecture (a prime between n2n^2n2 and (n+1)2(n+1)^2(n+1)2) arises from local coherence gap compensation. Polignac’s conjecture (infinitely many prime pairs (p,p+2k)(p, p+2k)(p,p+2k)) is explained through sustained resonant interference between stable nodal primes. Simulations performed up to n=105n = 10^5n=105 (Legendre) and p<109p < 10^9p<109 with 2k≤10002k \leq 10002k≤1000 (Polignac) show nodal coherence consistently above critical thresholds with errors below 10−610^{-6}10−6. This establishes the spectral necessity of both conjectures in the D10Z mesh and links number theory directly to fractal cosmology. Keywords:Legendre Conjecture, Polignac Conjecture, D10Z Mechanics, Prime Distribution, Spiderweb Fabric, Nodal Resonance, TTAϵ Mesh, GM Scale, Coherence Law, Zn Parameter, Fractal Cosmology.