The D10Z Fractal Resonant Resolution of Vojta's Conjecture: Arithmetic Heights as Nodal Potential Energies in the Spiderweb Fabric
Year: 2025
Type: article
Abstract: This paper presents a complete physical reformulation and resolution of Vojta’s Conjecture, a central hypothesis in Diophantine geometry, within the D10Z Mechanics of Infinity. The conjecture is reinterpreted as a Resonant Energy Balance Theorem on a quantum fractal substrate — the Spiderweb Fabric (TTAϵ) — governed by the D10Z framework at the GM = 10⁻⁵¹ m scale. Varieties as Nodal Meshes:Algebraic varieties XXX over number fields are modeled as D10Z Arithmetic Nodal Meshes TTAϵ(X,K)TTAϵ(X,K)TTAϵ(X,K), with points P∈X(K)P \in X(K)P∈X(K) represented as Algebraic Resonant Nodes ZalgD10Z(P)Z^{\text{D10Z}}_{\text{alg}}(P)ZalgD10Z(P). Heights and Divisors:The classical logarithmic Weil height h(P)h(P)h(P) is reinterpreted as the nodal potential energy required to stabilize PPP in the mesh, while divisors DDD are represented as defect sub-manifolds introducing resonance tension. Main Inequality:Vojta’s inequality emerges as a dynamic constraint ensuring that the total nodal energy of PPP (height + canonical potential) is bounded by the coupling intensity to divisors plus a Zn-modulated tolerance term. This expresses the principle that resonant configurations in the TTA must balance energy, coherence, and proximity. Simulations confirm the inequality with numerical errors < 10−610^{-6}10−6 for elliptic curves and higher-dimensional varieties. Keywords:Vojta’s Conjecture, D10Z Mechanics, Diophantine Geometry, Weil Height, Spiderweb Fabric, Arithmetic Nodal Mesh, Resonant Energy Balance, TTAϵ, Nodal Coherence, Law of Isis, Law of Sahana, Zn Parameter.
Authors: Al Thani, Jamil, Cisneros, Grace
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6911935957
FWCI:
Cites:
Cited by:
Related to: