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Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend theFundamental Theorem of $K$-theory from the category of simplicial rings to thecategory of $\mathbb{S}$-algebras. As an intermediate step, we prove theFundamental Theorem for simplicial rings appealing to recent results from thefirst author's thesis. This recovers as a special case the Fundamental Theoremfor the $K$-theory of spaces appearing inH\"uttemann-Klein-Vogell-Waldhausen-Williams.

A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras