ZEBTS: zero extremum of the topological boundary spectrum

This paper presents the ZEBTS (Zero‑Entropy Background Topological Structure) theoretical program — a unified operator‑spectral framework aimed at deriving space, time, matter, energy, and fundamental interactions from a single underlying spectral object, the flytron operator DχD_χ. The core hypothesis is that the complete physical reality R=(G,T,M,F,Q)R = (G, T, M, F, Q)R=(G,T,M,F,Q) can be reconstructed from the spectrum Spec⁡(Dχ)Spec(D_χ)Spec(Dχ) via a constructive functional R=Φ(Spec⁡(Dχ))R = Φ(Spec(D_χ))R=Φ(Spec(Dχ)), without introducing independent primary entities. The work develops a rigorous axiomatic basis (axioms ZA1–ZA8) that defines the spectral structure, existence conditions for geometry, mechanisms of dynamics, and criteria for observable reconstruction. It establishes a formal link between spectral data and geometric quantities: topology is encoded in the algebra AAA, while the metric is recovered via Connes’ distance formula d(x,y)=sup⁡{∣f(x)−f(y)∣:f∈A, ∥[Dχ,f]∥≤1}d(x,y) = \{ |f(x) - f(y)| : f ∈ A,\ \| [D_χ, f] \| ≤ 1 \}d(x,y)=sup{∣f(x)−f(y)∣:f∈A, ∥[Dχ,f]∥≤1}. Time is interpreted as the parameter of the heat flow U(t)=exp⁡(−tDχ2)U(t) = exp(-t D_χ^2)U(t)=exp(−tDχ2), and matter as stable excitations of the spectral modes of DχD_χ. A discrete version of the theory is constructed to enable numerical verification. It includes discrete spectral triplets (AN,HN,Dχ(N))(A_N, H_N, D_χ(N))(AN,HN,Dχ(N)), numerical protocols NP1–NP3, and diagnostic criteria D1–D4. Convergence results are proven: the discrete spectrum converges to the continuous one, spectral coefficients ak(N)a_k(N)ak(N) approximate their continuum counterparts with error bounds ∣ak(N)−ak∣≤CkN−βk|a_k(N) - a_k| ≤ C_k N-β_k∣ak(N)−ak∣≤CkN−βk, and diagnostic criteria remain stable under mesh refinement. Key physical limits are derived within the framework: General Relativity emerges as the low‑energy expansion of the spectral action Sspec=Tr⁡(f(Dχ2))Sspec = Tr(f(D_χ^2))Sspec=Tr(f(Dχ2)), yielding the Einstein–Hilbert action plus higher‑order curvature invariants; quantum phenomena (superposition, entanglement, interference, tunneling) follow directly from the linear and tensor structure of the Hilbert space and the spectral representation. Compared to existing unification programs, ZEBTS differs by its foundational architecture: space, time, and matter are not postulated independently but are derived from a single spectral source; the approach extends Connes’ spectral geometry to include the origin of matter, cosmology, and information; and it provides a numerically verifiable protocol with explicit error control. The paper outlines open problems and future directions, emphasizing the need for quantitative predictions and experimental tests.