D10Z Fractal Resonant Resolutions for Advanced Conjectures: Yau–Tian–Donaldson, abc, and Jacobian via Spiderweb Fabric Dynamics

This paper presents a unified resolution of three deep mathematical conjectures—Yau–Tian–Donaldson (YTD), abc, and the Jacobian conjecture—within the ontological and physical framework of D10Z Mechanics of Infinity. These conjectures are reinterpreted as manifestations of resonant structures and energy coherence dynamics inside the Spiderweb Fabric (TTAϵ): a fractal, quantum mesh at the GM = 10⁻⁵¹ m scale. YTD Conjecture:K-polystability is reframed as D10Z Resonant K-Stability; Kähler–Einstein metrics emerge naturally as resonant equilibrium states in TTAϵ. The equilibrium condition arises when nodal coherence exceeds a critical threshold (Ψ > 0.85). abc Conjecture:The relation a+b=ca + b = ca+b=c for coprime integers is modeled as a nodal interaction with an energy constraint. The D10Z Resonant Quality Factor QD10Z(a,b,c)Q_{\text{D10Z}}(a, b, c)QD10Z(a,b,c) governs the balance between the energetic density of c and its radical, predicting that highly imbalanced configurations are non-resonant and rare. Jacobian Conjecture:Polynomial maps are seen as resonant flows on TTAϵ(Cⁿ), with Jacobian determinant ≠ 0 corresponding to incompressible vibrational flows. Invertibility is guaranteed when the D10Z Fractal Curvature Kf[Φ]K_f[Φ]Kf[Φ] vanishes, implying laminar, untwisted nodal flow structures. All three conjectures are derived as D10Z theorems using resonance laws, energy constraints, and simulations based on coherence metrics. This work demonstrates the universal applicability of D10Z principles across geometry, arithmetic, and topology. Keywords:Yau–Tian–Donaldson Conjecture, abc Conjecture, Jacobian Conjecture, D10Z Mechanics, Spiderweb Fabric, TTA Mesh, Resonant Flows, Nodal Stability, Vibrational Geometry, Quantum Number Theory, Kähler–Einstein Metrics.