The Sigma-Displacement Law: A Geometric Proof of the Riemann Hypothesis
Description
This pre print introduces a novel experimental method for the validation of the Riemann Hypothesis, termed the "Inverse Falsification Test". We posit a model wherein prime numbers, when projected through a "lens" defined by the non-trivial zeros of the Riemann Zeta Function, form a 3D geometric structure in a space we call the "Riemann Vortex". In turn, this prime structure is utilized as a secondary lens to analyze the integrity of the zeros themselves. We demonstrate that a Riemann zero which satisfies the Hypothesis (with real part σ=0.5) projects to a coherent position within the Vortex, whereas a "false zero" (with σ≠0.5) undergoes a massive and measurable geometric displacement.
Finally, we establish that the sensitivity of this test is directly proportional to the magnitude of the zeros used in the lens. A comparative experiment shows that a lens constructed from high-energy zeros (magnitude ~10²¹), based on the computations of A. M. Odlyzko, is over 23 times more effective at detecting the falsification for 20-digit primes than a lens of low-energy zeros. Furthermore, we establish the fundamental law governing this displacement. A linear regression on the log-log plot reveals a near-perfect power-law relationship.
Contact:
andrespirolo@gmail.com
Related Work. Only on Zenodo.
Other preprints by this author:
The Chaotic Vortex Score: A Deterministic Knot Invariant with Applications to DNA Topology Classification
https://zenodo.org/records/17469726
The Chaotic Vortex Physical Descriptor: Separating Trivial and Non-Trivial Topological Regimes in Knotted Polymers
https://zenodo.org/records/17471374
The Consistency Peak Principle - Geometric Identification of the Chromatic Number via 3D Riemann-Vortex Mapping
https://zenodo.org/records/17403219
A Geometric Anomaly Detection Method for Locating Large Mersenne Prime Candidates
https://zenodo.org/records/17267849
The Signature of Chaos
https://zenodo.org/records/17239094
Zero-Vortex: A Geo-Arithmetic Paradigm for the Yang-Mills Mass Gap
https://zenodo.org/records/17204948
The Sigma-Displacement Law: A Geometric Proof of the Riemann Hypothesis
https://zenodo.org/records/17096978
Universal Geometric Signatures in 3-SAT
https://zenodo.org/records/17412980
A Geo-Arithmetic Duality: The Pirolo Vortex and a Computational Criterion for Algebraic Cycles
https://zenodo.org/records/17156451
The Sigma-Displacement Law: A Geometric Proof of the Riemann Hypothesis
https://zenodo.org/records/17096978
Contact: apirolo@abc.gob.ar
Notes (English)
Author's Note & Call for Collaboration:
As an independent researcher, my primary goal is to share these findings with the scientific community for review, criticism, and further development. This work has been made publicly available on Zenodo to ensure its permanent archival and citability.
I would be deeply grateful for any guidance or support from established members of the community in navigating this final procedural step. If you are qualified to endorse new authors on arXiv and find this work to be a serious contribution worthy of discussion, your endorsement would be invaluable.
The necessary information provided by arXiv is:
Endorsement Link: https://arxiv.org/auth/endorse?x=8W9JAY
Endorsement Code: 8W9JAY
For feedback, criticism, or collaboration, please feel free to contact me at: andrespirolo@gmail.com
Thank you for your time and consideration.
Files
A Geometric Proof of the Riemann Hypothesis--V2.0-1.pdf
Files
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Additional details
Dates
- Updated
-
2025-09-11Added Code Availability section, GitHub repository link, and formal references
Software
- Repository URL
- https://github.com/andydevok/Pirolo-Vortex