In the companion paper~\cite{ Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative $n$-ary $\Gamma$-semirings $(T,\Gamma)$ in terms of finitely generated projective $n$-ary $\Gamma$-modules and their automorphisms, and we identified the low K-groups $K_{ 0}(T,\Gamma)$ and $K_{ 1}(T,\Gamma)$ with appropriate Grothendieck and Whitehead groups. The present paper continues this programme by constructing and comparing several models for the higher algebraic K-theory of $(T,\Gamma)$. Starting from the Quillen-exact category $\mathcal{ C} := T\text{ -Mod}^{ \mathrm{ bi}}$ of bi-finite, slot-sensitive $n$-ary $\Gamma$-modules introduced earlier, we define the higher K-groups $K_{ n}(T,\Gamma)$ via Quillen's Q-construction~\cite{ Quillen73} on $\mathcal{ C}$ and via Waldhausen's $S_{ \bullet}$-construction~\cite{ Waldhausen85} on the Waldhausen category of bounded chain complexes in $\mathcal{ C}$. We prove a Gillet--Waldhausen type comparison theorem~\cite{ GilletGrayson87} showing that the resulting Quillen and Waldhausen K-theory spectra are canonically weakly equivalent. Using dg-enhancements and the derived category of quasi-coherent sheaves on the non-commutative spectrum $\operatorname{ Spec}_{ T}^{ \mathrm{ nc}}(T)$~\cite{ Gokavarapu_JRMS_2266}, we further identify these spectra with the K-theory of the small stable $\infty$-category of perfect complexes~\cite{ Thomason90}. As consequences, we obtain functoriality, localization, and excision sequences~\cite{ Weibel13}, and a derived Morita invariance statement for $K_{ n}(T,\Gamma)$~\cite{ Keller94}. These results show that algebraic K-theory of non-commutative $n$-ary $\Gamma$-semirings is a derived-geometric invariant of $\operatorname{ Spec}_{ \Gamma}^{ \mathrm{ nc}}(T)$ and reduce concrete computations to geometric d\'evissage and homological techniques developed in the earlier papers of the series.
Categories: math.RA math.KT