ZEBTS: An Axiomatic Framework for Spectral Arithmetic and the Riemann Hypothesis

@misc{mukha2026zebts,
  title = {{ZEBTS: An Axiomatic Framework for Spectral Arithmetic and the Riemann Hypothesis}},
  author = {Anatolii Mukha},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23438597},
  url = {https://zenodo.org/records/23438597},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - ZEBTS: An Axiomatic Framework for Spectral Arithmetic and the Riemann Hypothesis
AU  - Anatolii Mukha
PY  - 2026
DA  - 2026/10/17
PB  - Zenodo
DO  - 10.5281/zenodo.23438597
UR  - https://zenodo.org/records/23438597
AB  - This paper introduces and develops the ZEBTS framework (Law of the Unity of Being of the Triune Substance) as a novel axiomatic system designed to address the Riemann Hypothesis (RH) through the lens of spectral arithmetic. The central innovation of ZEBTS lies in its explicit unification of three interdependent domains—arithmetic structure, discrete geometry, and spectral theory—into a single, internally consistent mathematical architecture.  At the core of the framework is the arithmetic graph ΓZ\Gamma_ZΓZ, whose vertices correspond to prime numbers and whose edge weights are determined by arithmetic functions of the primes, specifically we∝(log⁡p)−2w_e \propto (\log p)^{-2}we∝(logp)−2. On this discrete structure, we define the self-adjoint operator HχH_\chiHχ, whose spectral properties are rigorously tied to the distribution of primes and the zeros of the Riemann zeta function. The axiomatic foundation of ZEBTS is established through a minimal set of axioms (G5‑A, G6, G6‑L’, etc.), each formulated to capture a fundamental aspect of the triune relationship between arithmetic, geometry, and spectrum. Crucially, all primitive objects are formally defined prior to the introduction of axioms, ensuring the logical minimality and internal consistency of the system.  Key theoretical developments include the construction of the spectral block of ZEBTS, comprising definitions of the spectral zeta function, spectral determinant, and heat function, along with the proof of the theorem on the abscissa of convergence. A trace formula of Gutzwiller type is derived, providing an explicit link between sums over primes and sums over the eigenvalues of HχH_\chiHχ. This formula is instrumental in controlling the error term in the prime number theorem and forms a critical step toward the RH proof.  Furthermore, we introduce a discrete analogue of Hodge theory tailored to the arithmetic graph ΓZ\Gamma_ZΓZ. In this setting, algebraic cycles are replaced by closed paths with unit weight product, and Hodge classes correspond to spectral subspaces localized on cluster classes. Axiom G6‑L’ emerges as the discrete counterpart to the classical Hodge conjecture, serving as the central open problem of the framework. Its resolution is shown to be equivalent to establishing the full spectral correspondence required for the RH proof within ZEBTS.  To bridge theory and empirical validation, we formulate a suite of numerical verification protocols (Protocols 4.1–4.3). These blueprints detail how to construct finite-dimensional approximations Hχ(N)H_\chi^{(N)}Hχ(N), compute nearest-neighbor level spacings to test for Gaussian Unitary Ensemble (GUE) statistics, estimate the spectral gap δ0(N)\delta_0(N)δ0(N), and analyze the localization properties of eigenfunctions with respect to arithmetic structures (e.g., smooth composite numbers). Expected numerical results are specified in quantitative terms, including tabulated predictions and plot descriptions, thereby providing falsifiable benchmarks for future computational studies.  Notably, the ZEBTS approach extends beyond RH, revealing conceptual connections to other major problems in mathematics and physics. The spectral gap δ0\delta_0δ0 of the operator HχH_\chiHχ is interpreted as an arithmetic analogue of the mass gap in quantum field theory, suggesting potential relevance to the Yang–Mills existence and mass gap problem. Previous work by the author on spectral structures in Yang–Mills theory (submitted to the Journal of Mathematical Physics, AIP) provides preliminary groundwork for this cross-disciplinary link.  The paper is structured to clearly delineate axiomatic assumptions, proven theorems, and proposed verification protocols. Section 2 lays the axiomatic groundwork; Section 3 develops the spectral block; Section 7 presents the trace formula; Section 8 introduces the discrete Hodge-theoretic interpretation; Section 10 details the numerical verification blueprints; Section 12 isolates and analyzes the central open problem G6‑L’; and the Conclusion synthesizes the results and assesses the framework’s novelty. This layered presentation ensures transparency and facilitates rigorous peer evaluation.  In summary, ZEBTS offers a constructive, axiomatic, and verifiable program for addressing the Riemann Hypothesis. By embedding arithmetic data into a geometrically structured spectral problem and integrating numerical validation directly into the theoretical design, the framework provides a clear pathway for further research and computational testing. The combination of strict axiomatics, deep theoretical connections, and concrete verification criteria positions ZEBTS as a distinct and promising approach within the landscape of RH research.
ER  -