Proof of the Triunity of Substance Based on the ZEBTS Framework
Abstract
This work presents ZEBTS‑SUPER, a new operator–spectral framework that unifies arithmetic, geometry, topology, and ontology into a single mathematically rigorous architecture. The theory introduces χ‑Absolute as the fundamental continuous substrate of reality, the Boundary (Flaitrons) as its discrete actualizations, and χ‑Reality as the spectral manifestation produced by the internal action of the Boundary operator Tx. This triadic structure — χ‑Absolute → Boundary → χ‑Reality — is not postulated but derived as a theorem from the spectral, analytic, and arithmetic mechanisms developed throughout the monograph. Sections 11 and 12 establish the complete analytic chain linking the axioms A1–A5, PT‑symmetry, the intertwining identity, the Euler‑product structure, the functional equation, growth order, zero density, and confinement. Numerical verification (T1–T6) confirms the spectral density, growth‑order gap, phase linearity, self‑dual geometry, and spacing structure. Analytically, the completed determinant D^(z) is shown (conditionally) to have growth order 1, zero density of Riemann–von Mangoldt type, and quadruple symmetry of zeros. Under the determinant‑level functional equation, all zeros are confined to the self‑dual circle, yielding the Riemann Hypothesis as a corollary of the spectral geometry. The global novelty of ZEBTS‑SUPER is structural. It provides the first mathematically rigorous model in which matter, space, time, and physical laws emerge from a single operator acting on χ‑Absolute. Flaitrons appear as minimal, topologically stable localizations of χ‑Absolute; χ‑Reality arises as the spectral output of Tx; and the Boundary created by χ‑Reality determines the next act of Tx, forming a closed, self‑deterministic, spectrally complete cycle. This establishes the Triunity of Substance as a theorem: χ‑Absolute → Boundary → χ‑Reality → new Boundary → …, a structure that simultaneously explains the origin of physical reality and reduces the Riemann Hypothesis to a single analytic condition. ZEBTS‑SUPER therefore constitutes a new fundamental theory of physics and mathematics. It unifies spectral geometry with arithmetic, resolves the geometric–arithmetic growth‑order gap, provides the first confinement theorem derived from symmetry and growth order, and demonstrates that RH is not an isolated conjecture but a consequence of the deeper ontological law — the Triunity of Substance.
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BibTeX
@misc{mukha2026proof,
title = {{Proof of the Triunity of Substance Based on the ZEBTS Framework}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22794111},
url = {https://doi.org/10.5281/zenodo.22794111},
note = {CC BY},
}
RIS
TY - UNPB
TI - Proof of the Triunity of Substance Based on the ZEBTS Framework
AU - Anatolii Mukha
PY - 2026
DA - 2026/09/16
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22794111
UR - https://doi.org/10.5281/zenodo.22794111
AB - This work presents ZEBTS‑SUPER, a new operator–spectral framework that unifies arithmetic, geometry, topology, and ontology into a single mathematically rigorous architecture. The theory introduces χ‑Absolute as the fundamental continuous substrate of reality, the Boundary (Flaitrons) as its discrete actualizations, and χ‑Reality as the spectral manifestation produced by the internal action of the Boundary operator Tx. This triadic structure — χ‑Absolute → Boundary → χ‑Reality — is not postulated but derived as a theorem from the spectral, analytic, and arithmetic mechanisms developed throughout the monograph. Sections 11 and 12 establish the complete analytic chain linking the axioms A1–A5, PT‑symmetry, the intertwining identity, the Euler‑product structure, the functional equation, growth order, zero density, and confinement. Numerical verification (T1–T6) confirms the spectral density, growth‑order gap, phase linearity, self‑dual geometry, and spacing structure. Analytically, the completed determinant D^(z) is shown (conditionally) to have growth order 1, zero density of Riemann–von Mangoldt type, and quadruple symmetry of zeros. Under the determinant‑level functional equation, all zeros are confined to the self‑dual circle, yielding the Riemann Hypothesis as a corollary of the spectral geometry. The global novelty of ZEBTS‑SUPER is structural. It provides the first mathematically rigorous model in which matter, space, time, and physical laws emerge from a single operator acting on χ‑Absolute. Flaitrons appear as minimal, topologically stable localizations of χ‑Absolute; χ‑Reality arises as the spectral output of Tx; and the Boundary created by χ‑Reality determines the next act of Tx, forming a closed, self‑deterministic, spectrally complete cycle. This establishes the Triunity of Substance as a theorem: χ‑Absolute → Boundary → χ‑Reality → new Boundary → …, a structure that simultaneously explains the origin of physical reality and reduces the Riemann Hypothesis to a single analytic condition. ZEBTS‑SUPER therefore constitutes a new fundamental theory of physics and mathematics. It unifies spectral geometry with arithmetic, resolves the geometric–arithmetic growth‑order gap, provides the first confinement theorem derived from symmetry and growth order, and demonstrates that RH is not an isolated conjecture but a consequence of the deeper ontological law — the Triunity of Substance.
ER -