ZEBTS: Discrete Geometry, Flytrons, and Emergent Spectra

@misc{mukha2026zebts,
  title = {{ZEBTS: Discrete Geometry, Flytrons, and Emergent Spectra}},
  author = {Anatolii Mukha},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23479930},
  url = {https://zenodo.org/records/23479930},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - ZEBTS: Discrete Geometry, Flytrons, and Emergent Spectra
AU  - Anatolii Mukha
PY  - 2026
DA  - 2026/10/19
PB  - Zenodo
DO  - 10.5281/zenodo.23479930
UR  - https://zenodo.org/records/23479930
AB  - ZEBTS: Discrete Geometry, Flytrons, and Emergent Spectra  This paper introduces the ZEBTS framework as a self‑contained formalism that constructs continuous spectral structures and geometric notions from a discrete, ontologically grounded network. The central innovation lies in the concept of the flytron—an elementary act of structuring the χ‑Boundary—which is elevated to the status of the primary object; all conventional mathematical and physical entities are derived as structural manifestations of the flytron network.  The discrete geometry of ZEBTS is built on a graph ZKZ_KZK whose vertices correspond to primes ppp and whose edges encode lll-adic adjacency rules. Each vertex ppp hosts a flytron Φp\Phi_pΦp, which defines the local gluing rule for its neighborhood and thereby determines the local topology. Cluster classes {PK}\{PK\}{PK} are defined as strata of entanglement across scales: small KKK corresponds to large‑scale, “classical” correlations, while large KKK encodes fine‑scale quantum fluctuations. This stratification yields a multiscale description that unifies arithmetic, topology, and correlation structure within a single discrete setup.  Spectra in ZEBTS are emergent: they arise from the collective dynamics of the flytron network rather than being imposed as external data. A self‑adjoint operator DFD_{\mathfrak{F}}DF is explicitly constructed on ZKZ_KZK, and its spectrum is shown to encode resonant modes of the network. The spectral identity det⁡reg(DF−z)=ξ(12+iz)\det_{\mathrm{reg}}(D_{\mathfrak{F}}-z) = \xi(\frac{1}{2}+iz)detreg(DF−z)=ξ(21+iz) is established and numerically verified, linking the discrete network to analytic properties of the zeta function. Importantly, this identity is not assumed but derived from the underlying discrete geometry and flytron dynamics, providing a concrete realization of the Hilbert–Pólya program within a computable framework.  Gravity and spacetime geometry are also emergent phenomena in ZEBTS. Spacetime emerges as a macroscopic manifestation of quantum entanglement between flytron cells, while gravity is identified with the collective behavior of a flytron condensate—a coherent state of strongly entangled cells. The condensate’s density generates an effective energy‑momentum tensor, leading to equations formally analogous to Einstein’s field equations, yet fully derived from spectral and correlation properties of the discrete network. This construction ensures compatibility with quantum physics: geometry is not fundamental but arises from quantum correlations, resolving the traditional tension between quantum mechanics and general relativity at the ontological level.  The axiomatic core of ZEBTS (axioms ZA1–ZA7) is reinterpreted in terms of flytron dynamics and associated with concrete analytical risks. Axiom L1 is introduced to guarantee the completeness of cluster coverage, ensuring that every flytron belongs to some cluster class. The rigidity theorem (Thm. 9.2) eliminates non‑physical deformations, while Lemma G6‑L′ serves as a universal reduction mechanism that translates global conjectures into local spectral conditions on cluster strata.  Numerical verification confirms the internal consistency of the framework. Computations on truncated graphs ZKZ_KZK demonstrate convergence of zero counts via direct spectral analysis and contour integration, with deviations below 1.7%1.7\%1.7% for Pmax⁡=5⋅103P_{\max} = 5\cdot10^3Pmax=5⋅103. Trace asymptotics are confirmed with relative errors under 0.5%0.5\%0.5%, and the behavior of cluster counts, barrier heights, and conductivity aligns with the predicted hierarchy. These results establish the reproducibility and computational verifiability of the model, even without high‑performance computing environments.  By grounding spectra, geometry, and gravity in a discrete, flytron‑based network, ZEBTS offers a unified, computationally accessible formalism where continuous structures emerge from discrete, ontologically well‑defined acts of structuring. The framework not only provides candidate operators and spectral identities but also introduces new calculable observables—such as condensate density, cluster entanglement strata, and discrete spectral gaps—that can be tested numerically or, potentially, inferred from physical measurements. This work lays the foundation for further analytical and experimental investigation into the deep connections between discrete geometry, spectral theory, and the quantum structure of spacetime.
ER  -