Finite-menu analytic slice families conjecture

Construct finitely many explicit analytic slice families (A_i,B_i): X_i -> SU(2)^2, each with a compact connected parameter space X_i and a commuting basepoint x_i^0 in X_i, such that for every hard positive-word difference w = u^{-1}v arising from distinct positive words with ab(u)=ab(v), there exist an index i and a parameter t in X_i satisfying tr(w(A_i(t),B_i(t))) ≤ 0, thereby ensuring a trace-zero witness on one of the slices by continuity.

Background

The authors introduce a slice-driven approach that converts algebraic invariants of positive-word differences into explicit low-dimensional families in SU(2)^2 on which traces can be computed. They provide several successful slice criteria but identify a residual super-degenerate class not covered by the current methods.

They formulate a conjectural completion target: a finite collection of explicit analytic slice families with commuting basepoints that collectively detect every hard positive-word difference by producing nonpositive trace somewhere on one of the slices, which implies an exact trace-zero witness via the intermediate value principle.