About the work
This manuscript presents a final conditional version of a proof of the Riemann Hypothesis, developed within the framework of S-finite adelic systems.
The approach is axiomatic and independent: starting from an abstract scale-flow system with discrete orbit lengths, we derive the prime structure (ℓᵥ = log qᵥ) and construct a canonical determinant D(s) without presupposing the Euler product or ζ(s).
The proof establishes:
D(s) is entire of order ≤ 1 and satisfies D(1−s) = D(s).
The zero measure of D coincides with that of Ξ(s) on a Paley–Wiener determining class with multiplicities (Appendix A, Theorem A.1).
Asymptotic normalization holds: lim₍σ→∞₎ log D(σ+it) = 0.
By Hadamard factorization, D(s) ≡ Ξ(s).
Consequently, all non-trivial zeros of ζ(s) lie on Re(s) = 1/2, subject to the conditional validity of the axioms.
Transparency and reproducibility are central:
All technical sketches of earlier drafts have been expanded into full Hardy-style proofs with explicit inequalities and references.
Numerical validations up to error 10⁻⁶ are included (Appendix C), with falsification tests confirming ℓᵥ = log qᵥ uniquely.
Complete code, data, and appendices are openly available in the GitHub repository and Zenodo record.
Status note: This is a final conditional version. It does not claim community validation but is offered with full transparency for rigorous expert review.
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Title A Complete Proof of the Riemann Hypothesis via S-Finite Adelic Systems (Final Conditional Version V4.1)
This manuscript presents a final conditional version of a proof of the Riemann Hypothesis, developed within the framework of S-finite adelic systems.
The approach is axiomatic and independent: starting from an abstract scale-flow system with discrete orbit lengths, we derive the prime structure (ℓᵥ = log qᵥ) and construct a canonical determinant D(s) without presupposing the Euler product or ζ(s).
The proof establishes:
D(s) is entire of order ≤ 1 and satisfies D(1−s) = D(s).
The zero measure of D coincides with that of Ξ(s) on a Paley–Wiener determining class with multiplicities (Appendix A, Theorem A.1).
Asymptotic normalization holds: lim₍σ→∞₎ log D(σ+it) = 0.
By Hadamard factorization, D(s) ≡ Ξ(s).
Consequently, all non-trivial zeros of ζ(s) lie on Re(s) = 1/2, subject to the conditional validity of the axioms.
Transparency and reproducibility are central:
All technical sketches of earlier drafts have been expanded into full Hardy-style proofs with explicit inequalities and references.
Numerical validations up to error 10⁻⁶ are included (Appendix C), with falsification tests confirming ℓᵥ = log qᵥ uniquely.
Complete code, data, and appendices are openly available in the GitHub repository and Zenodo record.
Status note: This is a final conditional version. It does not claim community validation but is offered with full transparency for rigorous expert review.
Work type Education, Informative
Tags we derive the prime structure (ℓᵥ = log qᵥ) and c, zeta-free construction of the canonical determina, the riemann hypothesis in full. an axiomatically
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Identifier 2509193115617
Entry date Sep 19, 2025, 5:17 PM UTC
License All rights reserved
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Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Sep 19, 2025.
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