ZEBTS: Cluster Classes, Discrete Geometry, and the Reduction of RH to Lemma G6‑L'

@misc{mukha2026zebts,
  title = {{ZEBTS: Cluster Classes, Discrete Geometry, and the Reduction of RH to Lemma G6‑L'}},
  author = {Anatolii Mukha},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23441115},
  url = {https://zenodo.org/records/23441115},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - ZEBTS: Cluster Classes, Discrete Geometry, and the Reduction of RH to Lemma G6‑L'
AU  - Anatolii Mukha
PY  - 2026
DA  - 2026/10/17
PB  - Zenodo
DO  - 10.5281/zenodo.23441115
UR  - https://zenodo.org/records/23441115
AB  - This paper develops the ZEBTS framework (An Axiomatic Framework for Spectral Arithmetic and the Riemann Hypothesis) as a constructive pathway to the Riemann Hypothesis (RH) by combining discrete geometry, spectral theory, and a tailored discrete Hodge formalism. The central achievement is the explicit reduction of RH to Lemma G6‑L’, a structural statement within the framework that links the arithmetic of prime numbers to the spectral properties of a rigorously defined operator through the geometry of cluster classes.We introduce the flaytron graph ΓZ\Gamma_ZΓZ, a prime-indexed discrete structure in which vertices correspond to primes p∈Pp \in \mathbb{P}p∈P and edges encode nearest-neighbor arithmetic relationships. On this graph, we define a weighted Hilbert space Hw=ℓ2(P,w)\mathcal{H}_w = \ell^2(\mathbb{P}, w)Hw=ℓ2(P,w) with weights wp=(log⁡p)−2w_p = (\log p)^{-2}wp=(logp)−2, chosen to ensure convergence of spectral traces and compatibility with the multiplicative structure of the integers. Within this space, we construct an explicit spectral operator HχH_\chiHχ with tridiagonal matrix representation governed by the logarithmic weights; its finite-dimensional approximations Hχ(N)H_\chi^{(N)}Hχ(N) are computable for large NNN (e.g., N≥104N \ge 10^4N≥104).A key innovation of ZEBTS is the introduction of cluster classes—equivalence classes of subgraphs on ΓZ\Gamma_ZΓZ that capture collective behaviors of primes over intervals and serve as discrete analogues of algebraic cycles. These classes are endowed with a precise geometric and algebraic structure via the axiomatic system (axioms FW, G5‑A', G6, S3, S4), which formalizes the geometry of ΓZ\Gamma_ZΓZ, the weight structure, and the stability of the spectral data. We show that cluster classes provide a natural basis for understanding spectral projections of HχH_\chiHχ, enabling a geometric interpretation of the distribution of its eigenvalues.  The main theoretical result is the reduction of RH to Lemma G6‑L’: the Riemann Hypothesis holds if and only if the subspaces spanned by cluster classes coincide with the spectral subspaces of HχH_\chiHχ associated with critical regions of the spectrum. This lemma transforms RH from an analytic problem about the zeros of ζ(s)\zeta(s)ζ(s) into a structural question about the alignment of discrete geometric data (cluster classes) and spectral data (eigenspaces of HχH_\chiHχ). We establish the necessary functional-analytic foundation for HχH_\chiHχ, including self-adjointness via adapted Kato–Rellich arguments, spectral gap estimates, and a trace formula connecting spectral sums to arithmetic quantities involving primes.To support empirical validation, we provide fully reproducible numerical protocols for computing spectral statistics of Hχ(N)H_\chi^{(N)}Hχ(N), including the pair correlation function R2(s)R_2(s)R2(s) and Dyson’s Δ3(L)\Delta_3(L)Δ3(L) statistic, and for comparing these results with Random Matrix Theory (GUE) predictions. Reference Python implementations are included in the appendices, ensuring independent verification of all numerical claims.Compared to prior approaches, ZEBTS offers a distinct constructive perspective: unlike the statistical methods of Montgomery and Odlyzko, it provides a deterministic operator generating the observed statistics; in contrast to the Berry–Keating heuristic, it replaces divergent continuous models with a well-defined discrete operator; and relative to Connes’ noncommutative framework, it operates on a simpler, explicitly computable graph structure while preserving deep arithmetic connections. Moreover, the framework reveals conceptual parallels with other Millennium Prize Problems, particularly through the role of the spectral gap in relation to the mass gap in Yang–Mills theory and the analogy between Lemma G6‑L’ and the Hodge conjecture. The paper is organized as follows: Section 2 presents the minimal axiomatic system; Section 3 establishes functional-analytic properties of HχH_\chiHχ; Section 4 develops spectral theory and the trace formula; Section 5 constructs the discrete Hodge theory and defines cluster classes rigorously; Section 6 proves the reduction of RH to Lemma G6‑L’; Section 7 refines the trace formula to connect spectral data with the prime counting function π(x)\pi(x)π(x); Section 8 details numerical verification protocols; Section 9 explores connections to other open problems; and Section 10 formulates open problems with explicit success criteria. Appendices contain technical derivations, auxiliary lemmas, and reference code. Our results demonstrate that the Riemann Hypothesis can be studied within a fully axiomatic, geometrically grounded, and computationally verifiable framework. The explicit construction of HχH_\chiHχ, the rigorous definition of cluster classes, and the availability of reproducible numerical tests position ZEBTS as a coherent and viable approach to one of mathematics’ most profound challenges.
ER  -