Spectral Realization of Riemann Zeta Zeros via the Adèlic Glushkov Operator and Informational Field Gradient Collapse.

Title: Spectral Realization of Riemann Zeta Zeros via the Adèlic Glushkov Operator and Informational Field Gradient Collapse. Background and Methodology: For over 160 years, the distribution of prime numbers has remained linked to the distribution of zeros of the Riemann zeta function ζ(s). This work introduces a novel approach rooted in "Field Ontology" and the "Kirlian-modulation of semantic invariants" method, bridging the gap between biological homeostasis laws and analytic number theory. Key Theoretical Framework: Adèlic Hilbert Space: The operator Ĥζ is defined on the ring of adèles AQ, providing a natural framework for automorphic forms and Hecke algebra integration. Operator Symmetry: We prove the functional equation Ĥζ ∘ FA = FA ∘ (I - Ĥζ), where FA is the unitary adèlic Fourier transform. This symmetry forces the spectrum to be invariant under the transformation s ↔ 1-s. Gradient Collapse & Stability: Utilizing the Glushkov Protocol 2.0, we introduce a non-linear dissipative term G(Ŝ) that ensures the convergence of eigenvalues to the point of maximum stability (U ≥ Ucrit). Resolvent Estimate: We establish the bound \|(s - Ĥζ)⁻¹\| ≤ C/|σ - 1/2|, demonstrating that any deviation from the critical line σ = 1/2 results in an infinite energy barrier in the limit N → ∞. Conclusion: The spectral proof concludes that the critical line is the only stable manifold for the system's resonances. Consequently, all non-trivial zeros of ζ(s) must lie on the line (s) = 1/2. Author Information: Oleg Glushkov, author of the Field Ontology theory and the Kirlian-modulation method. This work integrates 20 years of research in homeostasis, blood circulation evidence bases (Nikitin experiments), and digital simulations of Glushkov’s Saturation.