Exact Separation of Words via Trace Geometry

A basic question in the study of measure-once quantum finite automata is whether two distinct input words can be separated with certainty. The exact separation problem reduces to a trace-vanishing question in $SU(2)$. The main difficulty lies in the genuinely nonabelian regime, where $u$ and $v$ have the same abelianization. This paper develops a slice-driven framework that converts algebraic invariants of the word – prefix statistics, metabelian polynomials, and slope specializations – into explicit low-dimensional families in $SU(2{)}^2$ on which the trace-vanishing question can be analyzed effectively. A quadratic trace-deficit identity on a principal one-parameter family provides the main algebraic-to-geometric bridge. Building on this framework, the paper establishes three core certified slice criteria: a dihedral criterion, equivalently readable through a signed $a$-count; a quaternionic criterion; and a local one-row criterion. Together with a supplementary interior-point test and a binary-dihedral slice, these results sharply reduce the unresolved portion of the problem to a residual super-degenerate class, while also clarifying the limitations of certification strategies based only on finitely many finite-subgroup evaluations.