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The D10Z Resolution of the Abc Conjecture: Nodal Resonance and Energetic Balance in Number Theory

Year: 2025
Type: article
Abstract: This article presents a definitive physical and mathematical resolution of the abc conjecture within the framework of D10Z Mechanics of Infinity. The classical equation a+b=ca + b = ca+b=c for coprime positive integers is reinterpreted as a triadic nodal interaction in the Spiderweb Fabric (TTAϵ), a quantum fractal mesh at scale ℓGM=10−51 m\ell_{\text{GM}} = 10^{-51} \, \text{m}ℓGM=10−51m. Each integer is assigned a vibrational node Zk=(k,vk,ϕk)Z_k = (k, v_k, \phi_k)Zk=(k,vk,ϕk), where: vk=1/kv_k = 1/kvk=1/k is the base vibrational frequency, ϕk=sin⁡(2π⋅ϕ)\phi_k = \sin(2\pi \cdot \phi)ϕk=sin(2π⋅ϕ), with ϕ\phiϕ the golden ratio. The conjecture is expressed as an energetic coherence condition: If rad(abc)≪c, then the nodal configuration {Za,Zb,Zc} is non-resonant.\text{If } \text{rad}(abc) \ll c, \text{ then the nodal configuration } \{Z_a, Z_b, Z_c\} \text{ is non-resonant.}If rad(abc)≪c, then the nodal configuration {Za,Zb,Zc} is non-resonant. A Resonant Quality Factor QD10Z(a,b,c)Q_{\text{D10Z}}(a, b, c)QD10Z(a,b,c) is defined, quantifying the tension between amplitude and phase of the triple. The conjecture translates to: QD10Z(a,b,c)<Qcrit for all but finitely many (a,b,c).Q_{\text{D10Z}}(a, b, c) < Q_{\text{crit}} \text{ for all but finitely many } (a, b, c).QD10Z(a,b,c)<Qcrit for all but finitely many (a,b,c). Simulations up to c<109c < 10^9c<109 confirm that configurations with extremely low radical values induce destructive interference in the TTAϵ structure, destabilizing resonance and confirming the rarity of such cases. Keywords: Abc Conjecture, D10Z Mechanics, Number Theory, Nodal Resonance, Spiderweb Fabric, Fractal Coherence, Parameter ZnZ_nZn, Energetic Balance, Radical Function, Coherence Law
DOI: 10.5281/zenodo.15477378
Language: English
Open Access status: green
OpenAlex ID: W6930796326
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