Topological network at the Planck scale in the ZEBTS framework: flytrons, triune substance, and foundations of quantum physics

This paper presents the geometric and spectral foundations of the ZEBTS (Law of the Unity of the Triune Substance) framework, in which core quantum structures and the Yang–Mills mass gap emerge as rigorous consequences of discrete topology, APS‑type index theory, and noncommutative spectral geometry. We construct a sequence of APS‑triunities TN=(ΣN,GN,Eχ(N))T_N = (Σ_N, G_N, E_χ(N))TN=(ΣN,GN,Eχ(N)), where discrete connections and holonomies encode the dynamics of the Flytron — the elementary carrier of structure in ZEBTS. The spectral Flytronus operator OFly(N)=(DF(N))2OFly(N) = (D_F(N))^2OFly(N)=(DF(N))2, built from the discrete Dirac operator DF(N)D_F(N)DF(N), governs the spectral properties of the network. Its spectrum is shown to be discrete and bounded away from zero by the TC‑gap γTC>0γTC > 0γTC>0, which follows from positive curvature estimates and the stability of the APS index. From this geometric setup, we derive key quantum features without additional postulates: Noncommutativity of localization and momentum observables arises as a direct consequence of discrete parallel transport and graph boundary effects. Specifically, the commutator [ΠW,PN]≠0[Π_W, P_N] ≠ 0[ΠW,PN]≠0 encodes the incompatibility between spatial localization and covariant variation across the boundary ∂W∂ W∂W. Quantization is obtained as a finite‑dimensional spectral phenomenon: every finite APS‑triunity naturally supports quantized energy levels, since the spectrum σ(OFly(N))={λk(N)}k=1∣VN∣σ(OFly(N)) = \{λ_k(N)\}ₖ₌₁|V_N|σ(OFly(N))={λk(N)}k=1∣VN∣ consists entirely of eigenvalues in the finite‑dimensional Hilbert space HN=L2(VN)H_N = L^2(V_N)HN=L2(VN). Topological classification of eigenstates is established via cluster structure, homology, and APS‑index sectors, providing a mathematical basis for quantum labels. Each eigenstate ψk(N)ψ_k(N)ψk(N) carries topological invariants: a cluster class, a homology class, and an APS‑index sector, determined by the support of the state. Superposition follows directly from the linearity of the underlying Hilbert space HNH_NHN: any linear combination ψ=∑iciψiψ = ∑_i c_i ψ_iψ=∑iciψi of admissible states remains in HNH_NHN, reproducing the superposition principle without independent axioms. Geometric correlations, induced by nontrivial holonomies and curvature, are rigorously formulated in terms of the holonomy correlator Cuv(ψ)=⟨ψ,Uuvψ⟩Cᵤᵥ(ψ) = ψ, Uᵤᵥ ψ (ψ)=⟨ψ,Uuvψ⟩. Their dependence on curvature is demonstrated via a discrete Stokes‑type expansion Hol⁡(Γ)=I+iFN⋅Area(Γ)+O(δN4)Hol(Γ) = I + i F_N · Area(Γ) + O(δ_N^4)Hol(Γ)=I+iFN⋅Area(Γ)+O(δN4), showing that Cuv(ψ)Cᵤᵥ(ψ)Cuv(ψ) varies nontrivially with the enclosed curvature when FN≠0F_N ≠ 0FN≠0. The existence of a spectral gap γTC>0γTC > 0γTC>0 and its persistence in the continuum limit provide an analytical route to the Yang–Mills mass gap within the ZEBTS framework. Moreover, arithmetic structures (e.g., localization at prime scales, coefficients apa_pap) are geometrically realized as curvature perturbations RNarithR_NᵃʳⁱᵗʰRNarith, linking the model to analytic number theory and highlighting a potential role of the Riemann hypothesis in the stability of spectral barriers. Concrete small‑network examples (N=3N=3N=3, N=4N=4N=4) illustrate the emergence of these phenomena in explicit matrix form. For instance, in the N=3N=3N=3 triangular network with holonomies U12=eiαU₁₂ = eiαU12=eiα, U23=eiβU₂₃ = eiβU23=eiβ, U31=eiγU₃₁ = eiγU31=eiγ and cycle phase Φ=α+β+γΦ = α + β + γΦ=α+β+γ, the asymmetry of the spectrum of OFly(3)OFly⁽³⁾OFly(3) directly reflects the influence of curvature. Appendices provide notation, omitted proofs (including the positive curvature estimate and APS‑index stability), continuum correspondences, and computational guidelines. ZEBTS thus offers a unified, parameter‑free construction where matter, void, and quantum laws originate from the interplay of topology, spectra, and arithmetic. The framework bridges discrete geometry, index theory, and theoretical physics, suggesting new directions for research in quantum gravity, spectral geometry, and the foundations of quantum theory.