  The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups
of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the
$K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups
of $L$. It is shown here that a modified version of this result generalises to
strongly $\mathbb{Z}$-graded rings; rather than the algebraic $K$-groups of
$L$, the splitting involves groups related to the shift actions on the category
of $L$-modules coming from the graded structure. (These action are trivial in
the classical case). The nil groups are identified with the reduced $K$-theory
of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris
and localisation sequences are established.
