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Preprint . 2025
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2025
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2025
License: CC BY
Data sources: Datacite
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The Sigma-Displacement Law: A Geometric Proof of the Riemann Hypothesis

Authors: Pirolo, Andrés Sebastián;

The Sigma-Displacement Law: A Geometric Proof of the Riemann Hypothesis

Abstract

This paper 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

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. To facilitate a broader discussion among mathematicians and physicists, I am currently in the process of submitting this manuscript to the arXiv preprint server. The final barrier for an independent researcher is the endorsement requirement for the `math.NT` (Number Theory) category. 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.

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
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