Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants

Abstract: We establish a foundational homotopical framework for ternary $Γ$-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 $Γ$-parameter space. This framework provides a rigorous homological bridge to Nambu mechanics and absolute geometry over $\mathbb{F}_1$.