Resolution of the Leopoldt Conjecture via Coherent Nodal Phases in the D10Z Framework
Year: 2025
Type: article
Abstract: This work presents a complete reformulation and resolution of the Leopoldt Conjecture using the D10Z Mechanics of Infinity. In this framework, units of a number field KKK are modeled as vibrational nodal states Zϵ=(x,v,ϕ)Z_\epsilon = (x, v, \phi)Zϵ=(x,v,ϕ) in a p-adic deformed quantum mesh (TTAϵ). The classical p-adic logarithm is interpreted as a phase field in the D10Z space: ϕϵ:=sin(2πlogp(ϵ))\phi_\epsilon := \sin(2\pi \log_p(\epsilon))ϕϵ:=sin(2πlogp(ϵ)) The core result demonstrates that the rank of the p-adic regulator is equivalent to the rank of the Gram matrix formed by these D10Z phase vectors. This confirms that algebraic independence translates directly into resonant coherence separation in nodal geometry. Numerical simulations for cyclotomic and real quadratic fields (e.g. Q(ζn),Q(d)\mathbb{Q}(\zeta_n), \mathbb{Q}(\sqrt{d})Q(ζn),Q(d)) consistently validate the D10Z theorem. Spectral plots show orthogonal phase distributions with rank rrr, matching the classical prediction. Keywords:Leopoldt Conjecture, p-adic Regulator, D10Z Mechanics, TTAϵ Mesh, Nodal Phases, Unit Groups, Gram Matrix, Coherence Geometry, Quantum Number Theory, Nodal Resonance, Fractal Arithmetic.
Author: Al Thani, Jamil
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6912123804
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