The Operator–Spectral Framework of ZEBTS: A Unified Structure for Geometry, Arithmetic, and the Riemann Hypothesis
Abstract
ZEBTS is an operator-spectral framework in which all domains of reality — quantum, geometric, gravitational, cosmological, informational, and cognitive — are postulated to arise from a single self-adjoint elliptic operator T_chi acting on a compact 2-dimensional Riemannian manifold S2_chi (the chi-sphere). The mathematical apparatus is proven: the spectral theorem (von Neumann, 1929) guarantees self-adjointness and real discrete spectrum; the Rellich lemma guarantees compact resolvent; Weyl asymptotics, Agmon localization, Kato-Rellich perturbation theory, and heat-kernel spectral geometry provide the functional-analytic foundation. The physical interpretation is postulated (the chi-Principle): all phenomena correspond to spectral properties of T_chi, partial geometric reconstruction via spectral invariants, and boundary stability governed by a self-adjoint operator S_chi. This article introduces four new mathematical structures that strengthen the framework to publication-ready rigour: (1) a generalized trace formula linking spectral sums to geometric integrals; (2) a zeta-regularized determinant det_zeta(T_chi) with meromorphic continuation of the spectral zeta function zeta_chi(s); (3) a postulated determinant-zeta correspondence identifying the Riemann zeta function as a derived invariant of T_chi; (4) an arithmetic Frobenius structure linking spectral weights to multiplicative number theory. Variational spectral bounds, numerical protocols with guaranteed error estimates, and experimental falsifiability criteria are provided throughout. For the canonical chi-sphere, numerical verification confirms: Weyl asymptotics (lambda_n ~ C * n, R^2 > 0.9999), convergence of zeta_chi(s) for Re(s) > 1, heat-kernel recovery of scalar curvature (R = 2), and trace-formula volume-term dominance. All postulates are explicitly identified; the mathematical foundation is proven independently of them.
Related
- Riemann Hypothesis as a Physical Law: Operator‑Spectral Reconstruction of Time from Zeta Zero Dynamics
- Riemann Hypothesis as a χ‑∞ Boundary Law: Complete Operator‑Spectral Proof in ZEBTS‑∞
- A genus-two spectral curve of the Riemann Ξ-function: Real Structure and Certified Arithmetic Separation of Periods
- The two-step spectral locus of the octonionic structure operator
- Spectral Reconstruction of Lorentz Structure from the χ‑Operator An Operator‑Theoretic Resolution of Non‑Transitivity in Special Relativity
BibTeX
@misc{mukha2026operatorspectral,
title = {{The Operator–Spectral Framework of ZEBTS: A Unified Structure for Geometry, Arithmetic, and the Riemann Hypothesis}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22896277},
url = {https://doi.org/10.5281/zenodo.22896277},
note = {CC BY},
}
RIS
TY - UNPB
TI - The Operator–Spectral Framework of ZEBTS: A Unified Structure for Geometry, Arithmetic, and the Riemann Hypothesis
AU - Anatolii Mukha
PY - 2026
DA - 2026/09/22
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22896277
UR - https://doi.org/10.5281/zenodo.22896277
AB - ZEBTS is an operator-spectral framework in which all domains of reality — quantum, geometric, gravitational, cosmological, informational, and cognitive — are postulated to arise from a single self-adjoint elliptic operator T_chi acting on a compact 2-dimensional Riemannian manifold S2_chi (the chi-sphere). The mathematical apparatus is proven: the spectral theorem (von Neumann, 1929) guarantees self-adjointness and real discrete spectrum; the Rellich lemma guarantees compact resolvent; Weyl asymptotics, Agmon localization, Kato-Rellich perturbation theory, and heat-kernel spectral geometry provide the functional-analytic foundation. The physical interpretation is postulated (the chi-Principle): all phenomena correspond to spectral properties of T_chi, partial geometric reconstruction via spectral invariants, and boundary stability governed by a self-adjoint operator S_chi. This article introduces four new mathematical structures that strengthen the framework to publication-ready rigour: (1) a generalized trace formula linking spectral sums to geometric integrals; (2) a zeta-regularized determinant det_zeta(T_chi) with meromorphic continuation of the spectral zeta function zeta_chi(s); (3) a postulated determinant-zeta correspondence identifying the Riemann zeta function as a derived invariant of T_chi; (4) an arithmetic Frobenius structure linking spectral weights to multiplicative number theory. Variational spectral bounds, numerical protocols with guaranteed error estimates, and experimental falsifiability criteria are provided throughout. For the canonical chi-sphere, numerical verification confirms: Weyl asymptotics (lambda_n ~ C * n, R^2 > 0.9999), convergence of zeta_chi(s) for Re(s) > 1, heat-kernel recovery of scalar curvature (R = 2), and trace-formula volume-term dominance. All postulates are explicitly identified; the mathematical foundation is proven independently of them.
ER -