Fractal Resonant Interpretation of the Collatz Conjecture via the D10Z Framework
Year: 2025
Type: article
Abstract: 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
Author: Al Thani, Jamil
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6912144844
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