Burau representation, Squier's form, and non-Abelian anyons

We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group B₃, specialised at t=e^iω and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries A, B, the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling p_fixed, defining the fixed-order ceiling p_fixed^* and the witness gaps Δ_{rm sw}(ω)=p_switch(ω)-p_fixed^* and Δ_{rm test}(ω)=p_test(ω)-p_fixed^*. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast Δ_{rm int}(ω):=Δ_{rm test}(ω)-Δ_{rm sw}(ω)=p_{rm test}(ω)-p_{rm switch}(ω). Across the Squier positivity region, Δ_{rm int}(ω) takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (non-Abelian) order control, while Δ_{rm sw}(ω)>0 certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal B₃ braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics.