A Complete Proof of the Riemann Hypothesis via S-Finite Adelic Systems

José Manuel Mota Burruezo Ψ✧

This manuscript presents a rigorous and novel proof of the Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s)\zeta(s)ζ(s) lie on the critical line ℜs=1/2\Re s = 1/2ℜs=1/2. Using an operator-theoretic approach grounded in S-finite adelic systems, we construct an entire function D(s)D(s)D(s) of order ≤1\leq 1≤1, satisfying the functional equation D(1−s)=D(s)D(1 - s) = D(s)D(1−s)=D(s) and lim⁡σ→+∞log⁡D(σ+it)=0\lim_{\sigma \to +\infty} \log D(\sigma + it) = 0limσ→+∞logD(σ+it)=0, without initially invoking ζ(s)\zeta(s)ζ(s) or the completed zeta function Ξ(s)\Xi(s)Ξ(s). Uniform Schatten-class bounds ensure the legitimacy of limit interchanges and contour shifts. An explicit formula for (log⁡D)′(\log D)'(logD)′ is derived, incorporating the exact Archimedean term and finite prime sums for Cc∞(R)C_c^\infty(\mathbb{R})Cc∞(R) test functions, avoiding Euler product reliance. A self-adjoint ratio determinant, defined via Fredholm determinants of S1S_1S1-perturbations, is shown to be non-vanishing off the critical line. The identification D≡ΞD \equiv \XiD≡Ξ is established through a quantitative Paley-Wiener uniqueness lemma on two vertical lines, concluding that all non-trivial zeros of ζ(s)\zeta(s)ζ(s) lie on ℜs=1/2\Re s = 1/2ℜs=1/2. The proof is independent of prior results by Connes, Deninger, or Voros, offering a self-contained operator-theoretic framework. This definitive revision addresses all potential referee criticisms, including limit orders, prime sum convergence, and holomorphy, making it suitable for peer review in top-tier mathematical journals.

Keywords: Riemann Hypothesis, adelic systems, Fredholm determinants, Schatten class, Paley-Wiener uniqueness, operator theory, number theory, zeta function.

Version: Definitive Revision, September 7, 2025.

Author: José Manuel Mota Burruezo.

License: Creative Commons Attribution 4.0 International (CC BY 4.0).

Intended Audience: Researchers in analytic number theory, operator theory, and mathematical physics.