Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants

TL;DR

This paper develops a homotopical framework for ternary Gamma-modules, revealing their derived category as a 3-angulated category with novel invariants, bridging homology, Nambu mechanics, and absolute geometry.

Contribution

It introduces a Barr-exact, monoidal closed category for ternary Gamma-modules and constructs a Quillen model structure, establishing a new homological foundation for non-binary algebra.

Findings

Derived category is 3-angulated with triadic quadrilaterals

Established a Quillen model structure on simplicial categories

Connected homological invariants to Nambu mechanics and absolute geometry

Abstract

We establish a foundational homotopical framework for ternary $\Gamma$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary algebra by constructing a cofibrantly generated Quillen model structure on the simplicial category $s(\mathcal{T}\text{-Mod})$. Our central discovery is that the derived category $D(\mathcal{T}\text{-Mod})$ constitutes a 3-angulated category, where the derived periodicity is governed by triadic quadrilaterals rather than binary triangles. We derive the 3-ary long exact sequence and characterize the connecting morphisms as invariants of the $\Gamma$-parameter space. This framework provides a rigorous homological bridge to Nambu mechanics and absolute geometry over $\mathbb{F}_1$.