Journal of Modern Classical Physics & Quantum Neuroscience

Abstract

This article presents a rigorous construction of complex analogues of the Riemann zeta function at the points zeta(- 1/2 + ib) and zeta(- 2 + ib) The central contribution is the identification of zeta(- 2 + ib) with the wave function of a quantum gravitational field parametrised by b: when bi bi the field behaves as a smooth continuum (classical general relativity); when bi <b: the geometry becomes granular and discrete, merging gravitation with quantum physics. The trivial zero zeta(- 2) = 0 expresses the perfect gravitational neutrality of flat Minkowski space-time, and serves as the base point for the complex perturbation zeta(- 2) -> zeta(- 2 + ib) that encodes quantum metric fluctuations.

Fundamental novelty. The spectral values sz[1] are no longer taken from an empirical look-up table or an asymptotic fitting formula. They are now defined as the unique roots in (-1, 0) of the implicit algebraic equation: 0(853-2453+245-8)^(2/3) (24ks72ks-24ks) 4s with kst, t ∈ (1,100). This gives the entire spectral table a rigorous algebraic foundation, extensible to any T EN* without interpolation. Numerical results confirm msrrS1S30 (machine precision) and approx - 3.6 * 10 ^ - 5 for the Dirac-field analogue. This work builds directly on the framework developed in [1].

It develops exclusively a Third Alternative: the interpretation of the observable muon/electron mass ratio mu/me = 207 as a dressed value produced by the renormalization of a bare spectral ratio Rm by quantum vacuum interaction. The central algorithm (Algorithm F) computes the extremal values reYI_1 and reXI_1 of the ratio Re[(-1/2+ibi)]/Re[(-1/2+ibz)] across all spectral pairs (b1, b2) generated by the implicit algebraic root structure, for extended k-grids (k) up to 20 and beyond) and N up to 10%. The key results are: Rm(k <= 10) \approx 275.86, Rm(k <= 20) = 392.42 , and a systematic powerlaw dependence of reY1_1 on k max, consistent with renormalization-group flow toward a non-trivial asymptotic bare mass ratio. A rigorous comparison section establishes the structural concordance between this spectral-arithmetic renormalization and the Wilsonian renormalization group. Extended algorithms (Algorithms G and H) implement GPU-accelerated computation for k up to 50 and 1 up to 10, enabling the first numerical investigation of the transition in the Third Alternative framework.