The D10Z Resonant Resolution of the Erdős–Turán Discrepancy Conjecture: Modular Decoherence as Vibrational Imbalance in the TTA Mesh

This work provides a resonant-fractal reinterpretation and resolution of the Erdős–Turán Discrepancy Conjecture within the D10Z Mechanics of Infinity. Rather than relying on classical uniform distribution metrics, the approach maps each integer ana_nan onto a vibrational node Zn=(n,vn,ϕn)Z_n = (n, v_n, \phi_n)Zn=(n,vn,ϕn) in the Spiderweb Fabric (TTAϵ), a quantum fractal mesh defined at GM-scale =10−51 m= 10^{-51} \, \mathrm{m}=10−51m. The discrepancy across residue classes modulo qqq is reframed as a D10Z-Modular Discrepancy DD10Z(q,m)D_{D10Z}(q, m)DD10Z(q,m), measuring the energy imbalance of nodal vibrational signatures Φq(A,m)\Phi_q(A, m)Φq(A,m) across the cyclic group Zq\mathbb{Z}_qZq. The paper introduces a Resonant Bound Conjecture: for any non-trivially uniform sequence AAA, DD10Z(q,m)/log⁡q≥CD10ZD_{D10Z}(q, m) / \sqrt{\log q} \geq C_{D10Z}DD10Z(q,m)/logq≥CD10Z for infinitely many qqq, with CD10Z∼0.1C_{D10Z} \sim 0.1CD10Z∼0.1. Numerical simulations validate this prediction for sequences such as {n}\{n\}{n} and {n2}\{n^2\}{n2}, showing “resonant collapse zones” consistent with high classical discrepancy. The framework confirms that global resonant silence (perfect uniformity) is forbidden by the Law of Sahana, which ensures decoherence thresholds across modular projections. Keywords:Erdős–Turán Conjecture, Modular Discrepancy, D10Z Mechanics, Spiderweb Fabric, TTA Mesh, Nodal Resonance, Vibrational Imbalance, Zn Parameter, Phase Decoherence, Sahana Law.