Unified D10Z Resolution of Lemoine's Conjecture and the Modularity Theorem via Fractal Modular

This paper presents a unified resolution of Lemoine’s Conjecture and the Modularity Theorem within the D10Z Mechanics of Infinity framework. Odd integers and rational elliptic curves are modeled as resonant attractor structures in the Spiderweb Fabric (SWF), operating on the quantum mesh TTAϵ at scale GM = 10⁻⁵¹ m. Lemoine’s Conjecture is recast as a coherence condition, where odd integers project onto prime-based beat pairs (p + 2q) with high nodal coherence Ψcoh(p, q) > 0.6. The Modularity Theorem becomes a D10Z Fractal-Modular Equivalence, where elliptic curves and modular forms are matched through vibrational node signatures and a coherence-preserving morphism TD10Z. Simulations verify both results: For all odd integers up to 10⁹, a valid decomposition with Ψcoh > 0.6 is found. D10Z nodal spectra of elliptic curves match their modular counterparts with R² > 0.999. Keywords:Lemoine Conjecture, Modularity Theorem, D10Z Mechanics, Resonance, Spiderweb Fabric, Elliptic Curves, Modular Forms, Φ(Z), Ψcoh, TTAϵ, Zn Parameter.