Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings

Abstract

We develop a comparison, base-change, and descent framework for the algebraic $K$-theory of non-commutative $n$-ary $\Gamma$-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi-finite, slot-sensitive $\Gamma$-modules and perfect complexes, we construct functorial maps on $K$-theory induced by extension and restriction of scalars under explicit $\Gamma$-flatness hypotheses in the relevant positional slots. We prove derived Morita invariance (via tilting bimodule complexes), establish Beck-Chevalley type base-change for cartesian squares, and deduce a projection formula compatible with the multiplicative structure coming from positional tensor products. Passing to the non-commutative $\Gamma$-spectrum Spec${}_{\Gamma }^{\mathrm{nc}}(T)$, we show locality for perfect objects and derive Zariski hyperdescent for $\mathbb{K}(\mathrm{Perf})$, together with excision and localization sequences for closed immersions and fpqc descent for $\Gamma$-flat covers. Finally, we interpret $K_{\Gamma }(X)$ geometrically as the $K$-theory of the stable $\infty$-category of $\Gamma$-perfect complexes, describe its universal property in $\Gamma$-linear non-commutative motives, and record compatibility with cyclotomic and Chern-type trace maps.