Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and D\'evissage

Abstract

Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of $K$-theory in this $\Ga$-parametrised, slot-sensitive setting. For the exact/Waldhausen categories of finitely generated bi-positional $n$-ary $\Ga$-modules, perfect complexes in the derived category, and perfect quasi-coherent complexes on the non-commutative $\Ga$-spectrum $\SpecGnC{T}$, we prove Waldhausen Fibration and Additivity theorems and Quillen-type Localization for Serre and Waldhausen pairs. Under natural hypotheses on $\Ga$-stable filtrations we obtain d\'evissage and Approximation theorems, together with cofinality and Karoubi invariance, showing that idempotent completion does not change $K$-theory and that cofinal subcategories control $K_n$ in positive degrees. We further derive a Bass--Quillen fundamental triangle for polynomial extensions in the $n$-ary $\Ga$-context and prove nilpotent invariance for two-sided $\Ga$-ideals. In geometric terms, these results yield localization and Mayer--Vietoris sequences for the $K$-theory of $\Perf(X)$ on $X=\SpecGnC{T}$ and its admissible open covers. Altogether, the paper shows that the higher $K$-theory of non-commutative $n$-ary $\Ga$-semirings enjoys the same formal properties as in the classical ring and scheme cases, providing a robust foundation for subsequent computational and homotopy-theoretic applications.