Arithmetic of Finite Field 37: A Computational Framework for Gauge Symmetry and Five-Fold Operator Algebra

This collection presents a five-part mathematical framework demonstrating that continuous geometric and physical structures—specifically the non-abelian gauge symmetries of the Standard Model and the topology of Calabi-Yau manifolds—can emerge natively from the discrete, deterministic arithmetic of the prime field F37

Developed through an extended, computationally verified collaboration between human intuition and large language models, the framework establishes a rigorous bridge between finite-field combinatorics, representation theory, and arithmetic geometry.

The collection consists of one open computational inquiry and five formal papers:

- Computational Inquiry (Cover Letter): An invitation to the arithmetic geometry community to compute the Weil polynomial of the Tian-Yau manifold over F37, proposing a definitive test of the framework’s phenomenological mapping.
    
- Finite Field 37 I: (Cyclotomic Uniqueness): Proves the unique algebraic positioning of p=37 via the sixth cyclotomic polynomial, establishing the base constants of the operator lattice.
    
- Finite Field 37 II: (The Discrete Imaginary): Explores the native imaginary unit i(37) = 6, showing how it natively controls quarter-rotations, Gaussian factorizations, and the gauge coupling of a symmetric space decomposition without requiring complex extensions.
    
- Finite Field 37 III: (Generalized Fusion-Lie Correspondence): Demonstrates that discrete n-ic cyclotomic additive fusion tensors over the field naturally generate continuous special orthogonal Lie algebras, specifically generating so(6) at n=6.
    
- Finite Field 37 IV: (The Adelic Fusion Algebra): Resolves the "Commutator Anomaly" (how an abelian period field generates a non-abelian Lie algebra) by mapping the fusion matrices to the maximal endomorphism algebra of the supersingular Klein Quartic via Tate's Theorem, fully characterizing the associated Dedekind zeta function.
    
- The Antimirror Redux: Formalizes the affine dynamics of the field into a modular harmonic oscillator, proving that the composition of fundamental involutions generates the Frobenius group G666 and perfectly partitions the quadratic residues.
   
The mathematical framework was conceived by J. Hosain beginning in 2023. Computational exploration, proof verification, and writing assistance were provided by Claude (Anthropic) and Gemini (Google). All algebraic claims have been independently verified by computer algebra. The author (me) takes full responsibility for all conjectures and interpretations.

https://github.com/OminiTurd/five-fold-algebra
https://publish.obsidian.md/444-619