Spectral invariants and spectral calibration in the ZEBTS model: zeta-determinant, topological index, and mass gap

This paper presents the ZEBTS (Law of the Unity of the Triune Substance) framework as a unified spectral‑topological approach to fundamental physics, where key invariants are derived from the arithmetic properties of the Riemann xi‑function ξ(s)ξ(s)ξ(s). We establish a direct link between the spectral data of a specially constructed operator HχH_χ and the imaginary parts of the nontrivial zeros of ξ(s)ξ(s)ξ(s) via the Spectral Calibration Postulate (SP‑1). Within this setup, we rigorously demonstrate the equivalence of several conditions (SB ⇔ G4‑QF ⇔ TC‑Gap ⇔ HP‑Amp) that guarantee the existence of a spectral gap. This spectral gap is then identified with the physical Yang–Mills mass gap, yielding a parameter‑free determination of ΔΔΔ. Furthermore, we show that the Atiyah–Patodi–Singer (APS) index becomes a computable arithmetic‑spectral quantity, expressed through a regularized trace tied to the logarithmic derivative of ξ(1/2+it)ξ(1/2+it)ξ(1/2+it). The model’s defining feature is the absence of free parameters: all fundamental quantities are fixed by the spectrum determined through SP‑1 and the known asymptotics of the zeros of ξ(s)ξ(s)ξ(s). The paper outlines numerical verification strategies, generalizations to other gauge groups and dimensions, and the relationship between the consistency of ZEBTS and the Riemann Hypothesis.