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| Médium: | Recurso digital |
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Zenodo
2026
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| On-line přístup: | https://doi.org/10.5281/zenodo.20124564 |
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- <h1>Abstract</h1> <p>We develop a unified operator–topological framework for fundamental physics, formulated in terms of the ZEBTS triple</p> <p>Z = ( ĝ\{\mu\nu}, D̂, ρ̂ ),</p> <p>where ĝ\{\mu\nu} is an operator metric, D̂ is a χ‑topological operator with non‑trivial spectral structure, and ρ̂ is a physical state defining χ‑expectation values.</p> <p>Within this framework we construct χ‑geometry, including the χ‑connection, χ‑curvature, χ‑scalar curvature, and the χ‑Einstein operator, all defined on a common χ‑domain D\χ.</p> <p>We formulate the χ‑variational principle and derive the χ‑Einstein equation</p> <p>Ĝ\{\mu\nu} = T̂\_{\mu\nu},</p> <p>together with the χ‑topological field equation for D̂.</p> <p>A χ‑scaling analysis shows that the χ‑action is renormalizable under broad spectral and operator‑geometric conditions, establishing ultraviolet stability of the theory.</p> <p>A central result of the work is the emergence of a spectral dark matter term.</p> <p>The spectral matter Lagrangian</p> <p>L̂\{spec}(D̂) = ∫ f(λ) dÊ(λ)</p> <p>produces a positive χ‑stress tensor T̂^{spec}\{\mu\nu} that contributes to gravity but does not couple to ordinary matter, providing a geometric and spectral explanation for dark matter without introducing new fields.</p> <p>We prove that χ‑geometry reduces to classical differential geometry in the commutative limit, and that the χ‑Einstein equation reduces to the classical Einstein equation, ensuring recovery of General Relativity.</p> <p>In χ‑FLRW symmetry, the χ‑Einstein equation yields an effective χ‑Friedmann equation</p> <p>H²(t) = (8πG/3) ρ\{eff}(t) + Λ/3,</p> <p>where ρ\{eff}(t) includes the spectral dark matter component.</p> <p>Under mild spectral conditions ρ(λ) f(λ) ≤ C λ^{q} with q < 2, all χ‑curvature invariants remain finite as t → 0, providing a spectral mechanism for nonsingular cosmological evolution.</p> <p>We further develop a quantum χ‑field theory, including χ‑canonical commutation relations, χ‑Hamiltonian evolution, and quantum dynamics of the χ‑topological operator, showing that the χ‑action remains stable under quantum corrections.</p> <p>Taken together, these results establish ZEBTS as a mathematically rigorous, renormalizable, and physically predictive operator–topological framework in which geometry, topology, spectral structure, quantum dynamics, and cosmology are unified.</p> <p>The emergence of dark matter, the recovery of General Relativity, and the nonsingular behaviour of χ‑cosmology demonstrate that ZEBTS provides a compelling foundation for a new operator‑geometric approach to fundamental physics.</p> <p> </p>