The $k$-server conjecture, a longstanding open problem in theoretical computer science, has been proven true. The work function algorithm achieves a competitive ratio of $k$ on every metric space, resolving the conjecture. The proof introduces a novel algebraic representation of the work function as a matrix, where all feasible paths to reach a configuration are encoded.

In this representation, minimum and addition operations correspond to formal addition and multiplication. Each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival triggers an update via a change of basis and row replacement.

The amortized analysis relies on a potential function defined using a larger matrix whose coordinates are coordinate pairs from the original representation. This work confirms that deterministic online algorithms can optimally solve the $k$-server problem. The proof leverages deep connections between linear algebra and online computation, providing a complete and rigorous verification of the conjecture first posed decades ago.