Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra (2512.11102v1)

Abstract: In the companion paper~\cite{Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative $n$-ary $Γ$-semirings $(T,Γ)$ in terms of finitely generated projective $n$-ary $Γ$-modules and their automorphisms, and we identified the low K-groups $K_{0}(T,Γ)$ and $K_{1}(T,Γ)$ 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 $(T,Γ)$. Starting from the Quillen-exact category $\mathcal{C} := T\text{-Mod}^{\mathrm{bi}}$ of bi-finite, slot-sensitive $n$-ary $Γ$-modules introduced earlier, we define the higher K-groups $K_{n}(T,Γ)$ 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 $K{n}(T,Γ)$~\cite{Keller94}. These results show that algebraic K-theory of non-commutative $n$-ary $Γ$-semirings is a derived-geometric invariant of $\operatorname{Spec}_Γ^{\mathrm{nc}}(T)$ and reduce concrete computations to geometric dévissage and homological techniques developed in the earlier papers of the series.