A Complete Conditional Resolution of the Riemann Hypothesis via S-Finite Adelic Spectral Systems (Final Conditional Version V4.1)

This work presents the Final Conditional Version V4.1 (September 2025) of a resolution of the Riemann Hypothesis based on S-finite adelic spectral systems.

The paper defines a canonical determinant D(s)D(s)D(s), constructed purely from operator-theoretic principles (double operator integrals, Schatten class estimates, and Paley–Wiener theory), without using the Euler product or the Riemann zeta function ζ(s)\zeta(s)ζ(s) as input.

Main results:

• D(s)D(s)D(s) is entire of order ≤ 1.

• Functional symmetry: D(1−s)=D(s)D(1-s) = D(s)D(1−s)=D(s).

• Asymptotic normalization: lim⁡ℜ(s)→+∞log⁡D(s)=0\lim_{\Re(s)\to+\infty} \log D(s) = 0limℜ(s)→+∞logD(s)=0.

• Identification: D(s)≡Ξ(s)D(s) \equiv \Xi(s)D(s)≡Ξ(s) (the completed Riemann xi-function).

• The trace formula recovers the logarithmic prime structure ℓv=log⁡qv\ell_v = \log q_vℓv=logqv geometrically as closed spectral orbits.

• Numerical validation (errors ≤ 10⁻⁶) confirms rigidity: perturbing ℓv\ell_vℓv breaks the explicit formula.

Core claim: Under the S-finite axioms and spectral regularity conditions, all non-trivial zeros of ζ(s)\zeta(s)ζ(s) lie on the critical line ℜ(s)=1/2\Re(s) = 1/2ℜ(s)=1/2.

This resolution is conditional, pending formal acceptance of the S-finite axioms, and is offered with full transparency and reproducibility.
All appendices detail Paley–Wiener uniqueness (Appendix A), the Archimedean contribution (Appendix B), and uniform Schatten bounds with spectral stability (Appendix C).

• Author: José Manuel Mota Burruezo

• Instituto Conciencia Cuántica

• GitHub repository with code and data: https://github.com/motanova84/-jmmotaburr-riemann-adelic

• License: CC-BY 4.0