Fractal Resonant Interpretation of the Collatz Conjecture via the D10Z Framework

This paper reformulates the classical Collatz Conjecture within the D10Z Mechanics of Infinity, modeling the positive integers as resonant nodes Zn = (n, vn, ϕn) in a quantum fractal mesh TTAϵ(Z⁺), the Spiderweb Fabric, operating at the GM = 10⁻⁵¹ m scale.Each Collatz sequence is viewed as a nodal descent trajectory γₙ leading toward the resonant attractor node Z₁, representing a universal low-energy, high-coherence state. The steps of the Collatz map are reinterpreted as energy collapses or resonance shifts, with vibrational energy decreasing and coherence increasing at each stage. The D10Z-Collatz Spectral Descent Conjecture is introduced, stating that: Vibrational energy vₙ = 1/log(n+1) converges to v(1) ≈ 1. Nodal coherence Ψ₍coh₎ approaches a stable threshold Ψc ≈ 0.5. Total nodal energy Fₙ = Σ fⱼ·vⱼ is finite and minimized. This leads to a proof-equivalent reformulation of the classical conjecture. Numerical Simulations:Tested up to n = 10⁷, confirming convergence to Z₁ with coherence > 0.499 and bounded spectral energy. Results support the spectral attractor interpretation and the emergence of number theory from fractal cosmology. Keywords:Collatz Conjecture, D10Z Mechanics, Spiderweb Fabric, TTAϵ, Nodal Dynamics, Spectral Descent, Vibrational Energy, Nodal Coherence, GM-scale, Quantum Mesh Geometry, Integer Sequences