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 $B_3$, specialised at $t=e^{i\omega }$ 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 $p_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}$, defining the fixed-order ceiling $p_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}^{\ast }$ and the witness gaps ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{w}}(\omega )=p_{\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}}(\omega )-p_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}^{\ast }$ and ${\mathrm{\Delta }}_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}(\omega )=p_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}(\omega )-p_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}^{\ast }$. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast ${\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}(\omega ):={\mathrm{\Delta }}_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}(\omega )-{\mathrm{\Delta }}_{\mathrm{s}\mathrm{w}}(\omega )=p_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}(\omega )-p_{\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}}(\omega )$. Across the Squier positivity region, ${\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}(\omega )$ takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (non-Abelian) order control, while ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{w}}(\omega )>0$ certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control that reproduces the characteristic interference pattern expected from a \emph{Gedankenexperiment} in anyonic statistics.