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Abstract
We present a rigorous operator-theoretic resolution to the Hilbert-Pólya conjecture, the classical Riemann Hypothesis (RH), and the Generalized Riemann Hypothesis (GRH) within the framework of Zero-Energy Bound State Triune Substance (ZEBTS) theory. We construct a one-dimensional Dirac-type operator Dχ acting on L^2(R, C^2) with a continuous hyperbolic mass potential V_0(u) = 2 cosh(u) and a singular prime delta-comb potential V_p(u) positioned at logarithms of prime numbers u_p = ln p. We prove that D_chi is strictly self-adjoint, unbounded, and possesses a purely discrete, real spectrum {gamma_n}. By evaluating the fundamental matrix solution across the prime jump discontinuities, we establish an exact equivalence between the spectral trace formula of D_chi and the Riemann-Weil explicit formula, identifying the spectrum Spec(D_chi) identically with the imaginary parts of the non-trivial zeros of the Riemann zeta function s_n = 1/2 + i * gamma_n. Extension to Dirichlet L-functions L(s, chi_q) is achieved via phase-twisted potential operators D_(chi, chi_q) preserving PT-symmetry and self-adjointness, proving GRH. Finally, we establish the second quantization of the field Psi(u, t) in 1+1 dimensions, demonstrating that the prime distribution governs quantum vacuum fluctuations and Casimir energy densities at the Planck scale. This unifies number theory, quantum field theory, and non-commutative geometry under the foundational Law of Unity of Being of the Triune Substance.
