The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it. Our proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal costs correspond to addition and multiplication of formal expressions, and each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival updates the representation via a change of basis and row replacement. The amortized analysis is based on a potential function defined in terms of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.
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