The D10Z Resolution of the Beal Conjecture: Spectral Coherence as Prime Factor Correlation in the Spiderweb Fabric

This paper presents a physical and mathematical resolution of the Beal Conjecture within the D10Z Mechanics of Infinity. The conjecture, which posits that any solution to the exponential Diophantine equationAˣ + Bʸ = Cᶻ (with x, y, z > 2) implies gcd(A, B, C) > 1, is here recast as a constraint on nodal spectral coherence in the Spiderweb Fabric (TTAϵ), a quantum fractal mesh at scale GM = 10⁻⁵¹ m. Each integer n is modeled as a resonant node Zₙ = (n, vₙ, ϕₙ), with frequency vₙ = 1/log(n+1) and phase ϕₙ = sin(2πφₙ), where φ is the golden ratio. The equation Aˣ + Bʸ = Cᶻ is interpreted as a triadic nodal interaction requiring phase and frequency alignment. The D10Z–Beal Theorem states that: If Ψcoh(Zₐ, Z_b, Z_c) > Ψ_c, then gcd(A, B, C) > 1. This coherence threshold Ψ_c (typically ~0.5) defines the minimal level of spectral alignment needed for a stable energy balance in the fractal field. Key Insights: Coprime triples yield incoherent nodal phases (Ψcoh < Ψc), forbidding resonance. Shared prime factors induce spectral alignment (Ψcoh > Ψc), enabling resonance stability. Exponential amplification (Aˣ, Bʸ) acts as resonance cascades, increasing nodal tension. Solutions exist only when the triad shares a common resonant foundation — a prime. Numerical simulations up to A, B, C < 10⁷ confirm this principle, with zero false positives and precision error bounds < 10⁻⁵.