Zariski Topology and Cohomology for Commutative Ternary Gamma Semirings

This paper develops the algebraic foundation required to build a Zariski-type geometry for \emph{commutative ternary $\mathrm{\Gamma }$-semirings}, where multiplication is an inherently triadic, multi-parametric interaction $(a,b,c,\gamma )\mapsto \{abc{\}}_{\gamma }$. Rather than treating triadic multiplication as an optional variation of binary algebra, we adopt it as an \emph{algebraic necessity} for modeling systems whose elementary interactions are intrinsically three-body and whose operational modes are indexed by parameters $\mathrm{\Gamma }$. We construct the prime spectrum ${\text{Spec}}_{\mathrm{\Gamma }}(T)$ and its Zariski topology, prove functoriality, and build the structure sheaf ${\text{Ocal}}_{{\text{Spec}}_{\mathrm{\Gamma }}(T)}$ via local fraction descriptions that must simultaneously respect triadic associativity and the sheaf gluing axioms. A key technical point is ensuring that local representations by ternary-parametric fractions glue uniquely, despite the absence of a binary product and despite the parameter dependence of the multiplication law. We then define sheaves of $\mathrm{\Gamma }$-modules, quasi-coherent sheaves associated to algebraic modules, and the cohomology groups $H^i(X,\text{Fcal})$ as derived functors of global sections. Finally, we give a concrete finite \emph{structural example} (a ternary $\mathrm{\Gamma }$-version of $\mathbb{Z}\mathrm{/}n\mathbb{Z}$) and compute its $\mathrm{\Gamma }$-spectrum explicitly, thereby exhibiting nontrivial spectral behavior in a fully finite setting.