We study the fully dynamic exact distance problem in unweighted directed graphs, where the graph undergoes edge insertions and deletions, and the distance between two query vertices must be reported. Under the combinatorial Boolean matrix multiplication conjecture, any subquadratic-time algorithm for this problem must use algebraic techniques. All prior work uses the polynomial matrix inverse as its algebraic tool, but within subquadratic time, this technique can only maintain distances up to a small-polynomial hop bound. So larger distances must be reassembled from short-range pieces by a combinatorial layer. This hybrid algebraic-combinatorial construction has led to a substantial slowdown for dynamic distances compared to other dynamic graph problems relying on algebraic techniques. We give the first subquadratic-time dynamic distance algorithm built on dynamically maintaining a different algebraic object than the matrix inverse. By maintaining the Frobenius form $A=KCK^{-1}$ of a dynamic generic (i.e., cyclic) matrix $A$, we can query all the powers $A^\ell = K C^\ell K^{-1}$ efficiently. For adjacency matrix~$A$, the smallest $\ell$ with $(A^\ell)_{s,t}\neq 0$ is the $st$-distance in the associated graph. So, unlike dynamic matrix inverse, we get distance information directly without hop truncation, and we no longer need to perform any combinatorial reassembly. This purely algebraic approach to dynamic distances leads to the fastest dynamic distance algorithm with $O(n^{1.58})$ update time. The earlier use of the Frobenius form maintained the dense change of basis $K$ explicitly and required $\tilde O (n^2)$ time per update [KS23]. Our key observation is that this change of basis is highly structured, which lets us maintain it implicitly. This gives the first subquadratic dynamic algorithm for the Frobenius normal form and characteristic polynomial for generic matrices.

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