A Multi-Scale Spectral Framework for General Relativity, Quantum Mechanics, and Classical Nonlinear Dynamics
Abstract
We establish a comprehensive, non-perturbative mathematical framework that unifies the three foundational pillars of theoretical physics:
1. The Einstein Field Equations of General Relativity: G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
2. The Schrödinger Wave Equation of Quantum Mechanics: i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi
3. The Governing Equation of Classical Nonlinear Dynamics: \frac{d^2 \mathbf{r}}{dt^2} + \mathbf{f}\left(\frac{d\mathbf{r}}{dt}, \mathbf{r}, t\right) = 0
We prove that these three formulations arise as exact, asymptotic projections of a single underlying master spectral system: the derived noncommutative spectral scheme \mathcal{U} \equiv \operatorname{Spec}(\mathcal{A}_{\text{eff}}) defined over the global adèle class space \mathbb{X}_{\mathbb{Q}} \equiv \mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^\times, governed by the four-sector coordinate Dirac operator \mathbf{D} = \mathbf{D}_M + \mathbf{D}_L + \mathbf{D}_G + \mathbf{D}_F on the zero-mean adèlic Hilbert space \mathcal{H}_{\mathbb{A}} = L_0^2(\mathbb{X}_{\mathbb{Q}}, d\mu).
The architectural synthesis resolves the foundational obstructions of quantum gravity, macroscopic cat-state superpositions, dissipation, and continuum divergence.
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BibTeX
@misc{tudor2026multiscale,
title = {{A Multi-Scale Spectral Framework for General Relativity, Quantum Mechanics, and Classical Nonlinear Dynamics}},
author = {August Tudor},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23515229},
url = {https://zenodo.org/records/23515229},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Multi-Scale Spectral Framework for General Relativity, Quantum Mechanics, and Classical Nonlinear Dynamics
AU - August Tudor
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23515229
UR - https://zenodo.org/records/23515229
AB - We establish a comprehensive, non-perturbative mathematical framework that unifies the three foundational pillars of theoretical physics: 1. The Einstein Field Equations of General Relativity: G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} 2. The Schrödinger Wave Equation of Quantum Mechanics: i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi 3. The Governing Equation of Classical Nonlinear Dynamics: \frac{d^2 \mathbf{r}}{dt^2} + \mathbf{f}\left(\frac{d\mathbf{r}}{dt}, \mathbf{r}, t\right) = 0 We prove that these three formulations arise as exact, asymptotic projections of a single underlying master spectral system: the derived noncommutative spectral scheme \mathcal{U} \equiv \operatorname{Spec}(\mathcal{A}_{\text{eff}}) defined over the global adèle class space \mathbb{X}_{\mathbb{Q}} \equiv \mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^\times, governed by the four-sector coordinate Dirac operator \mathbf{D} = \mathbf{D}_M + \mathbf{D}_L + \mathbf{D}_G + \mathbf{D}_F on the zero-mean adèlic Hilbert space \mathcal{H}_{\mathbb{A}} = L_0^2(\mathbb{X}_{\mathbb{Q}}, d\mu). The architectural synthesis resolves the foundational obstructions of quantum gravity, macroscopic cat-state superpositions, dissipation, and continuum divergence.
ER -