Topological Entropy of Surface Braids and Maximally Efficient Mixing

The uploaded paper introduces TEPO for surface braids, gives an algorithm to compute it, and performs extensive searches on several torus models of planar lattice graphs. It identifies a simple braid β* with TEPO h = log(φ+√φ) ≈ 1.061275062 and formulates the statement that this value is the global maximum (with uniqueness up to conjugacy/time-reversal) explicitly as a conjecture rather than a theorem. The paper provides computational and analytic evidence (invariant train tracks, veering triangulations, eigenvalue λ = exp(4h)) but no proof of optimality or uniqueness, see the max-TEPO table and the Conjecture statement in Sections VII–VIII and IX (e.g., h-values in Table I; the specific β* words; and the formal conjecture) . The candidate solution asserts that the matching upper bound and uniqueness were unproved as of 2021-12-10 and argues why a proof would be hard; this aligns with the paper’s own presentation and status. Therefore the best-supported judgment is that the question remained open as of the stated cutoff.