χ‑Spectral Geometry of Reality: Operator‑Geometric Origin of Spacetime, Curvature, and Gravity
Abstract
This work presents a complete operator‑spectral reconstruction of fundamental physics, establishing χ‑Reality as a fully finite, ultrametric, non‑geometric physical framework. Within the ZEBTS‑SUPER 9.0 architecture, spacetime, matter, interactions, quantum fields, gravity, cosmology, thermodynamics, and causal structure emerge as macroscopic projections of a single bounded operator algebra Aχ = {Φχ, Tχ, Ord(Tχ), Bχ, Aχ} acting on the ultrametric substrate Mχ. The spectral phase operator Φχ generates spatial geometry; the causal ordering operator Ord(Tχ) defines temporal direction and global hyperbolicity; the boundary tension functional Bχ governs gravitational and cosmological dynamics; and autopoietic transitions Aχ produce geodesic motion, thermodynamic flow, and structural evolution. We prove that χ‑geodesics are autopoietic spectral trajectories Λ(k) → Λ(k+1), whose continuum projection yields the classical Levi‑Civita geodesic equation uμ ∇μ uν = 0 in the low‑tension limit. χ‑spectral horizons arise as operator bifurcations at Bχ = Bχ(crit), eliminating all geometric singularities and ensuring global geodesic completeness. χ‑Spectral Cosmology introduces a double‑exponential expansion law aχ(k) = a0 exp(exp(−λp τχ)), resolving the Big Bang singularity, deriving the CMB temperature, dark energy density, cosmological constant Λχ = λ137^(−4), and large‑scale cosmic structure as spectral invariants. χ‑Spectral Thermodynamics defines temperature Tχ = Tr(Πk Ord(Tχ)), entropy Sχ = ln(dim HΛ(k)), and free energy Fχ = Eχ − Tχ Sχ, proving strict finiteness and eliminating all thermodynamic divergences. χ‑Spectral Quantum Field Theory constructs χ‑fields Φχ(k), χ‑quanta, χ‑propagators, and χ‑vacuum as bounded operators, yielding ultraviolet‑finite propagators Gχ(k,m) ≤ C² p^(−|k−m|), finite vacuum energy, and renormalization‑free dynamics. χ‑Spectral Unification derives U(1), SU(2), and SU(3) gauge sectors from prime‑mode automorphisms Aut(Pχ(p)) for p = 2, 3, 5, while gravity emerges from the operator triad (Bχ, Ord(Tχ), Φχ). In the continuum limit η → 0, Δk → 0, the Standard Model and Einstein gravity arise as macroscopic projections of the unified operator algebra Aχ. The resulting χ‑Reality framework constitutes the first fully finite, self‑consistent, operator‑spectral theory of fundamental physics, eliminating all ultraviolet divergences, infrared pathologies, geometric singularities, causal paradoxes, and thermodynamic inconsistencies, while providing a unified origin for spacetime, matter, interactions, and cosmology.
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BibTeX
@misc{mukha2026χspectral,
title = {{χ‑Spectral Geometry of Reality: Operator‑Geometric Origin of Spacetime, Curvature, and Gravity}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.21744121},
url = {https://doi.org/10.5281/zenodo.21744121},
note = {CC BY},
}
RIS
TY - UNPB
TI - χ‑Spectral Geometry of Reality: Operator‑Geometric Origin of Spacetime, Curvature, and Gravity
AU - Anatolii Mukha
PY - 2026
DA - 2026/08/01
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21744121
UR - https://doi.org/10.5281/zenodo.21744121
AB - This work presents a complete operator‑spectral reconstruction of fundamental physics, establishing χ‑Reality as a fully finite, ultrametric, non‑geometric physical framework. Within the ZEBTS‑SUPER 9.0 architecture, spacetime, matter, interactions, quantum fields, gravity, cosmology, thermodynamics, and causal structure emerge as macroscopic projections of a single bounded operator algebra Aχ = {Φχ, Tχ, Ord(Tχ), Bχ, Aχ} acting on the ultrametric substrate Mχ. The spectral phase operator Φχ generates spatial geometry; the causal ordering operator Ord(Tχ) defines temporal direction and global hyperbolicity; the boundary tension functional Bχ governs gravitational and cosmological dynamics; and autopoietic transitions Aχ produce geodesic motion, thermodynamic flow, and structural evolution. We prove that χ‑geodesics are autopoietic spectral trajectories Λ(k) → Λ(k+1), whose continuum projection yields the classical Levi‑Civita geodesic equation uμ ∇μ uν = 0 in the low‑tension limit. χ‑spectral horizons arise as operator bifurcations at Bχ = Bχ(crit), eliminating all geometric singularities and ensuring global geodesic completeness. χ‑Spectral Cosmology introduces a double‑exponential expansion law aχ(k) = a0 exp(exp(−λp τχ)), resolving the Big Bang singularity, deriving the CMB temperature, dark energy density, cosmological constant Λχ = λ137^(−4), and large‑scale cosmic structure as spectral invariants. χ‑Spectral Thermodynamics defines temperature Tχ = Tr(Πk Ord(Tχ)), entropy Sχ = ln(dim HΛ(k)), and free energy Fχ = Eχ − Tχ Sχ, proving strict finiteness and eliminating all thermodynamic divergences. χ‑Spectral Quantum Field Theory constructs χ‑fields Φχ(k), χ‑quanta, χ‑propagators, and χ‑vacuum as bounded operators, yielding ultraviolet‑finite propagators Gχ(k,m) ≤ C² p^(−|k−m|), finite vacuum energy, and renormalization‑free dynamics. χ‑Spectral Unification derives U(1), SU(2), and SU(3) gauge sectors from prime‑mode automorphisms Aut(Pχ(p)) for p = 2, 3, 5, while gravity emerges from the operator triad (Bχ, Ord(Tχ), Φχ). In the continuum limit η → 0, Δk → 0, the Standard Model and Einstein gravity arise as macroscopic projections of the unified operator algebra Aχ. The resulting χ‑Reality framework constitutes the first fully finite, self‑consistent, operator‑spectral theory of fundamental physics, eliminating all ultraviolet divergences, infrared pathologies, geometric singularities, causal paradoxes, and thermodynamic inconsistencies, while providing a unified origin for spacetime, matter, interactions, and cosmology.
ER -