Quadratic bounds for uncompletable words and matrix mortality

Abstract

Every finite nonempty incomplete uniquely decipherable code with maximum word length $k$ has an uncompletable word of length at most $4k^2-3k$. The bound is independent of the number of codewords and their total length. Deleting a complete codeword cycle gives a finite path-counting identity; Kraft equality then supplies a short word of deficient compressed mass. Cyclic averaging and padding turn it into an uncompletable word. Conditional expectation makes the construction polynomial-time and also decides completeness. First-return words extend the bound to mortal families of nonnegative integer $n\times n$ matrices with joint spectral radius at most one, provided every strongly connected component has a vertex meeting every cycle. Such a family has a zero product of length at most $4n^2-3n$. A binary partial deterministic family with $2k-1$ states has shortest zero product of length $k^2+k-1$, establishing the optimal quadratic order. The bounds and the explicit-code algorithm, including its polynomial work bound, are proved in Lean.