A Proof of the Artin Conjecture from the First Principles of Organized Complexity.
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2025
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| author | Okolo, Hanyelichukwu Paul |
| author_facet | Okolo, Hanyelichukwu Paul |
| contents | <div> <div> <div> <p>The Artin Conjecture on the entirety of non-trivial Artin L-functions is a central problem in algebraic number theory. This paper presents a complete proof of this conjecture derived from the Organized Complexity (OC) framework. We demon- strate that the conjecture is a necessary consequence of the framework’s physical realization of the Langlands correspondence. Within this model, a Galois representation ρ corresponds to a stable geometric pattern on the Harmonic Divisor Fan (HDF) lattice, while its Langlands dual, an automorphic form π, corresponds to the lattice’s unique stable vibrational mode. The framework’s Axiom of Dynamic Stability necessitates the identity of their associated L-functions, L(s, ρ) = L(s, π). Because the stability axiom also requires the automorphic L-function L(s,π) to be entire, representing a physically stable system free of runaway resonances, the Artin L-function L(s, ρ) must also be entire. This grounds the analytic properties of Artin L-functions in the fundamental requirement for a stable, coherent physical reality.</p> </div> </div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16003137 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Proof of the Artin Conjecture from the First Principles of Organized Complexity. Okolo, Hanyelichukwu Paul Artin Conjecture L-functions L-function <div> <div> <div> <p>The Artin Conjecture on the entirety of non-trivial Artin L-functions is a central problem in algebraic number theory. This paper presents a complete proof of this conjecture derived from the Organized Complexity (OC) framework. We demon- strate that the conjecture is a necessary consequence of the framework’s physical realization of the Langlands correspondence. Within this model, a Galois representation ρ corresponds to a stable geometric pattern on the Harmonic Divisor Fan (HDF) lattice, while its Langlands dual, an automorphic form π, corresponds to the lattice’s unique stable vibrational mode. The framework’s Axiom of Dynamic Stability necessitates the identity of their associated L-functions, L(s, ρ) = L(s, π). Because the stability axiom also requires the automorphic L-function L(s,π) to be entire, representing a physically stable system free of runaway resonances, the Artin L-function L(s, ρ) must also be entire. This grounds the analytic properties of Artin L-functions in the fundamental requirement for a stable, coherent physical reality.</p> </div> </div> </div> |
| title | A Proof of the Artin Conjecture from the First Principles of Organized Complexity. |
| topic | Artin Conjecture L-functions L-function |
| url | https://doi.org/10.5281/zenodo.16003137 |