Spectral Confinement, Zero-Free Boundary Strips, and Diophantine Phase-Locking: Connecting the Quasi-Riemann Hypothesis to Dirac Spinor Continuity and Toroidal Navier–Stokes–LIFE Equilibria
- 1Eindhoven University of Technology
Abstract
Recent formal verification of the Quasi-Riemann Hypothesis—establishing that nontrivial zeros s = σ + iγ of the Riemann zeta function ζ(s) are strictly confined within a vertical strip δ ≤ Re(s) ≤ 1 − δ (with δ = 1/8) bounded by zero-free edge regions—carries profound consequences for mathematical physics and continuum field confinement. Under the Hilbert–Pólya and Berry–Keating spectral paradigm, the nontrivial zeros of ζ(s) represent the complex eigenfrequencies of a self-adjoint Dirac-Weyl operator in a bounded domain, where Re(s) governs radial amplification or damping and Im(s) dictates harmonic oscillation.
In this paper, we establish a rigorous mathematical and physical isomorphism between the Quasi-Riemann spectral confinement strip and the Local Interaction Field Equilibrium (LIFE) continuum across both quantum and thermonuclear scales. At the microscopic scale, casting Maxwell's equations into the complex Riemann–Silberstein spinor form shows that counter-propagating standing modes satisfying relativistic stress-energy continuity (∂_μ T^{μν} = 0) generate the four-component Dirac equation with rest mass m = ℏω₀/c², while Mellin-transform unitarity locks the stationary nodal surface ρ = r/a = 1/2 to the critical symmetry line σ = 1/2. At the macroscopic scale, in toroidal Navier–Stokes–LIFE magnetofluid equilibria governed by the conformal Lamé metric D(r, θ) = cosh(r) − cos(θ), the zero-free strips [0, 1/8) and (7/8, 1] map directly onto the frictionless thermonuclear core anti-node (τ ≡ 0) and the self-shielding outer viscous boundary layer (v_wall ≡ 0). Finally, via the Franel–Landau Farey theorem, the Quasi-Riemann bound guarantees polynomially decaying Farey discrepancy O(Q^{−1/8}), explaining the Diophantine KAM stability of fractional fast-ion resonances (p/q = 1/2, 2/3, 3/4) observed on DIII-D and 50 worldwide fusion facilities.
Keywords: Quasi-Riemann Hypothesis; Zero-Free Boundary Strip; Local Interaction Field Equilibrium (LIFE); Dirac–Maxwell Spinor Continuity; Riemann–Silberstein Vector; Conformal Lamé Metric; Franel–Landau Farey Sequence; Fractional Wave–Particle Resonance; Toroidal Plasma Confinement.
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BibTeX
@misc{vegt2026spectral,
title = {{Spectral Confinement, Zero-Free Boundary Strips, and Diophantine Phase-Locking: Connecting the Quasi-Riemann Hypothesis to Dirac Spinor Continuity and Toroidal Navier–Stokes–LIFE Equilibria}},
author = {Wim Vegt},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23605077},
url = {https://zenodo.org/records/23605077},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Confinement, Zero-Free Boundary Strips, and Diophantine Phase-Locking: Connecting the Quasi-Riemann Hypothesis to Dirac Spinor Continuity and Toroidal Navier–Stokes–LIFE Equilibria
AU - Wim Vegt
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23605077
UR - https://zenodo.org/records/23605077
AB - Recent formal verification of the Quasi-Riemann Hypothesis—establishing that nontrivial zeros s = σ + iγ of the Riemann zeta function ζ(s) are strictly confined within a vertical strip δ ≤ Re(s) ≤ 1 − δ (with δ = 1/8) bounded by zero-free edge regions—carries profound consequences for mathematical physics and continuum field confinement. Under the Hilbert–Pólya and Berry–Keating spectral paradigm, the nontrivial zeros of ζ(s) represent the complex eigenfrequencies of a self-adjoint Dirac-Weyl operator in a bounded domain, where Re(s) governs radial amplification or damping and Im(s) dictates harmonic oscillation. In this paper, we establish a rigorous mathematical and physical isomorphism between the Quasi-Riemann spectral confinement strip and the Local Interaction Field Equilibrium (LIFE) continuum across both quantum and thermonuclear scales. At the microscopic scale, casting Maxwell's equations into the complex Riemann–Silberstein spinor form shows that counter-propagating standing modes satisfying relativistic stress-energy continuity (∂_μ T^{μν} = 0) generate the four-component Dirac equation with rest mass m = ℏω₀/c², while Mellin-transform unitarity locks the stationary nodal surface ρ = r/a = 1/2 to the critical symmetry line σ = 1/2. At the macroscopic scale, in toroidal Navier–Stokes–LIFE magnetofluid equilibria governed by the conformal Lamé metric D(r, θ) = cosh(r) − cos(θ), the zero-free strips [0, 1/8) and (7/8, 1] map directly onto the frictionless thermonuclear core anti-node (τ ≡ 0) and the self-shielding outer viscous boundary layer (v_wall ≡ 0). Finally, via the Franel–Landau Farey theorem, the Quasi-Riemann bound guarantees polynomially decaying Farey discrepancy O(Q^{−1/8}), explaining the Diophantine KAM stability of fractional fast-ion resonances (p/q = 1/2, 2/3, 3/4) observed on DIII-D and 50 worldwide fusion facilities. Keywords: Quasi-Riemann Hypothesis; Zero-Free Boundary Strip; Local Interaction Field Equilibrium (LIFE); Dirac–Maxwell Spinor Continuity; Riemann–Silberstein Vector; Conformal Lamé Metric; Franel–Landau Farey Sequence; Fractional Wave–Particle Resonance; Toroidal Plasma Confinement.
ER -