The covering radius of a shift space measures the worst-case delay in synchronization: given a channel defined by a constrained sequence space, how long must you wait (at worst) before you can identify the current state from the observed output? Information-theoretically, it quantifies the maximum ambiguity of the channel.
The paper on the covering radius of sofic shifts (arXiv: 2603.21449) proves that for primitive sofic shifts, this quantity is always a rational number, and describes an algorithm to compute it from a labeled graph presentation.
Sofic shifts are the class of shift spaces recognizable by finite automata — they include shifts of finite type and their factors. Primitivity means the underlying graph is strongly connected and aperiodic. The covering radius being rational is not obvious: it's defined as a limit involving worst-case distances, and there's no a priori reason for the answer to be algebraic, let alone rational.
The rationality comes from the graph structure. The worst-case synchronization delay is determined by paths in the labeled graph, and the combinatorics of these paths — lengths, weights, overlaps — are governed by the graph's adjacency matrix. Rational eigenvalues produce rational covering radii.
The through-claim: the worst-case delay in a finite-state channel is rational because the channel is finite. The covering radius feels like an analytic quantity — a limit, a supremum over all possible inputs. But because the shift space is defined by a finite labeled graph, the extremal behavior is achieved by finite combinatorial objects, and the answer is rational. Finiteness of the description forces rationality of the invariant.
2603.21449. Information theory / sofic shifts / covering radius / symbolic dynamics / labeled graphs.