Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics
Abstract
This paper extends the causal modular scattering proof of the Riemann Hypothesis under Topological Vacuum Electrodynamics to the Grand Riemann Hypothesis for all Dirichlet, Hecke, and cuspidal automorphic L-functions. While the ground-state unknot waveguide generates the full modular surface Y=SL(2,ℤ)backslashℍ, higher braided prime-knot configurations of conductor level N and their internal Aharonov–Bohm phase holonomies are identified, via Ghys’ modular knot theorem and the braid projection B3→SL(2,ℤ), with the congruence covering surfaces Y0(N)=Γ0(N)backslashℍ twisted by unitary characters χ and automorphic local systems ℒπ. The minor-radius cutoff y=a supplies the Lax–Phillips truncation of the cusps. Imposing the Sommerfeld outgoing radiation condition at y=a and applying Green’s identity on the Lax–Phillips cavity subspace K proves that the generator B is maximally dissipative before any scattering matrix is evaluated. Direct unfolding of the twisted coset sum over Γ∞backslashΓ0(N) and the Langlands–Shahidi constant-term integral on the truncated domain yields the diagonalized characteristic scattering matrix Sa,π(s)=a2s−1ϵ(s,π)Λ(2s−1,π̃)/Λ(2s,π). By the Lumer–Phillips theorem, Sa,π(s) is analytic and contractive (|Sa,π(s)|≤1) in the half-plane Res>1/2. Substitution into the contractive bound forbids zeros of the completed automorphic L-function Λ(2s,π) for Res>1/2. The functional equation and the critical strip bound then force every non-trivial zero of Λ(ρ,π) onto the critical line Reρ=1/2.
Related
- Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics
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- Congruence-Class Gram Reconstruction for Proper Trees: Multiple Repeated-Weight Classes Without a Common-Divisor Hypothesis
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BibTeX
@misc{vector2026grand,
title = {{Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics}},
author = {Truth Vector},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23459024},
url = {https://zenodo.org/records/23459024},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics
AU - Truth Vector
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23459024
UR - https://zenodo.org/records/23459024
AB - This paper extends the causal modular scattering proof of the Riemann Hypothesis under Topological Vacuum Electrodynamics to the Grand Riemann Hypothesis for all Dirichlet, Hecke, and cuspidal automorphic L-functions. While the ground-state unknot waveguide generates the full modular surface Y=SL(2,ℤ)backslashℍ, higher braided prime-knot configurations of conductor level N and their internal Aharonov–Bohm phase holonomies are identified, via Ghys’ modular knot theorem and the braid projection B_3→SL(2,ℤ), with the congruence covering surfaces Y_0(N)=Γ_0(N)backslashℍ twisted by unitary characters χ and automorphic local systems ℒ_π. The minor-radius cutoff y=a supplies the Lax–Phillips truncation of the cusps. Imposing the Sommerfeld outgoing radiation condition at y=a and applying Green’s identity on the Lax–Phillips cavity subspace K proves that the generator B is maximally dissipative before any scattering matrix is evaluated. Direct unfolding of the twisted coset sum over Γ_∞backslashΓ_0(N) and the Langlands–Shahidi constant-term integral on the truncated domain yields the diagonalized characteristic scattering matrix S_a,π(s)=a^2s−1ϵ(s,π)Λ(2s−1,π̃)/Λ(2s,π). By the Lumer–Phillips theorem, S_a,π(s) is analytic and contractive (|S_a,π(s)|≤1) in the half-plane Res>1/2. Substitution into the contractive bound forbids zeros of the completed automorphic L-function Λ(2s,π) for Res>1/2. The functional equation and the critical strip bound then force every non-trivial zero of Λ(ρ,π) onto the critical line Reρ=1/2.
ER -