Topological Operator Geometry of ZEBTS: Renormalizability, Quantum Consistency, and the Spectral Theory of Dark Matter
Abstract
The article develops the complete topological operator geometry of ZEBTS‑SUPER and establishes a fully renormalizable, anomaly‑free, modular‑invariant operator‑spectral cosmology based on the universal χ‑operator Dχ acting over the compact vacuum S³. The framework demonstrates that all fundamental physical phenomena — geometry, curvature, mass, interaction, expansion, acceleration, dark‑matter effects, and the elimination of Λ — arise from the spectral invariants, boundary structures, and commutator algebra of Dχ. Recent experimental results obtained by the Korean research group (2026) provide the first independent confirmation of χ‑Cosmology. Their measurements reveal spectral dilation consistent with the χ‑redshift law 1 + z = |λ_src| / |λ_obs|, boundary‑driven acceleration matching qχ < 0 for Kχ < 0 and Tχ,ren > 0, and high‑ordinate spectral confinement reproducing the χ‑spectral dark‑matter density ρ_dark(v) = Σ μχ |t_n| |ψ_n(v)|². These results validate the operator‑spectral origin of cosmic expansion and dark‑sector phenomena. The χ‑operator geometry constructed in this work is based on the discrete spectrum λ_n = 1/2 + i t_n, the χ‑spectral mass relation mχ(n) = μχ |t_n|, the χ‑distance dχ(v, w), and the χ‑curvature Kχ(v) = a₂(Dχ)(v). All ultraviolet divergences are removed by Seeley–DeWitt subtraction, yielding ζχ,ren(−1) = 0 and the exact elimination of the cosmological constant Λ. The operator algebra Aχ, generated by Dχ, Mf, dχ f, Rχ(f, g), and Oχ, is shown to be compact‑stable and SL(2,Z)‑modular invariant. The resulting cosmology is purely spectral: expansion arises from χ‑operator dilation, acceleration from χ‑boundary tension and negative curvature, and dark matter from high‑ordinate spectral projections. No additional fields, potentials, metrics, or vacuum energies are required. The theory is fully self‑consistent, renormalizable, and observationally supported. This work completes the operator‑spectral reconstruction of physical reality and establishes ZEBTS‑SUPER as a unified, mathematically rigorous, experimentally confirmed Theory of Everything based on topological operator geometry and χ‑spectral dynamics.
Related
- χ‑Spectral Geometry of Cosmological Mass Renormalization: Operator‑Topological Reconstruction of the Perimeter Institute "Dark Force" Effect
- Operator‑Spectral Geometry of Rotational Gravity: χ‑Modular Curvature, Spectral Flow, and the Precision Lense–Thirring Measurement of LARES‑2
- χ‑Spectral Geometry of Reality: Operator‑Geometric Origin of Spacetime, Curvature, and Gravity
- Superluminal Phenomena in Modern Physics: An Operator‑Topological Reconstruction in the ZEBTS‑SUPER Framework
- Superluminal Gravitational Waves: Operator‑Topological Reconstruction in the ZEBTS‑SUPER Framework
BibTeX
@misc{mukha2026topological,
title = {{Topological Operator Geometry of ZEBTS: Renormalizability, Quantum Consistency, and the Spectral Theory of Dark Matter}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.21788278},
url = {https://doi.org/10.5281/zenodo.21788278},
note = {CC BY},
}
RIS
TY - UNPB
TI - Topological Operator Geometry of ZEBTS: Renormalizability, Quantum Consistency, and the Spectral Theory of Dark Matter
AU - Anatolii Mukha
PY - 2026
DA - 2026/08/04
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21788278
UR - https://doi.org/10.5281/zenodo.21788278
AB - The article develops the complete topological operator geometry of ZEBTS‑SUPER and establishes a fully renormalizable, anomaly‑free, modular‑invariant operator‑spectral cosmology based on the universal χ‑operator Dχ acting over the compact vacuum S³. The framework demonstrates that all fundamental physical phenomena — geometry, curvature, mass, interaction, expansion, acceleration, dark‑matter effects, and the elimination of Λ — arise from the spectral invariants, boundary structures, and commutator algebra of Dχ. Recent experimental results obtained by the Korean research group (2026) provide the first independent confirmation of χ‑Cosmology. Their measurements reveal spectral dilation consistent with the χ‑redshift law 1 + z = |λ_src| / |λ_obs|, boundary‑driven acceleration matching qχ < 0 for Kχ < 0 and Tχ,ren > 0, and high‑ordinate spectral confinement reproducing the χ‑spectral dark‑matter density ρ_dark(v) = Σ μχ |t_n| |ψ_n(v)|². These results validate the operator‑spectral origin of cosmic expansion and dark‑sector phenomena. The χ‑operator geometry constructed in this work is based on the discrete spectrum λ_n = 1/2 + i t_n, the χ‑spectral mass relation mχ(n) = μχ |t_n|, the χ‑distance dχ(v, w), and the χ‑curvature Kχ(v) = a₂(Dχ)(v). All ultraviolet divergences are removed by Seeley–DeWitt subtraction, yielding ζχ,ren(−1) = 0 and the exact elimination of the cosmological constant Λ. The operator algebra Aχ, generated by Dχ, Mf, dχ f, Rχ(f, g), and Oχ, is shown to be compact‑stable and SL(2,Z)‑modular invariant. The resulting cosmology is purely spectral: expansion arises from χ‑operator dilation, acceleration from χ‑boundary tension and negative curvature, and dark matter from high‑ordinate spectral projections. No additional fields, potentials, metrics, or vacuum energies are required. The theory is fully self‑consistent, renormalizable, and observationally supported. This work completes the operator‑spectral reconstruction of physical reality and establishes ZEBTS‑SUPER as a unified, mathematically rigorous, experimentally confirmed Theory of Everything based on topological operator geometry and χ‑spectral dynamics.
ER -