Fixed SU(2) Fibonacci braid gate alphabet

Use the fixed chirality and fusion-tree basis of the three-$\tau$, total-$\tau$ Fibonacci space from Three-Fibonacci-anyon qubit matrices. There $B_1=\mathrm{diag}(1,r)$, $B_2=F{B}_{1}F$, and $r=e^{-3\pi i\mathrm{/}5}$. Put $z=e^{3\pi i\mathrm{/}10}$ and $G_{\mathrm{F}\mathrm{i}\mathrm{b}}=\{z{B}_1,z{B}_2,(z{B}_1{)}^{-1},(z{B}_2{)}^{-1}\}\subset SU(2)\mathrm{.}$ The determinants are one because $\det B_j=r$ and $z^{2}r=1$; the matrices are unitary by the cited matrix verification. The resulting package is inverse-closed, and its generated subgroup is dense under the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption by The generated normalized Fibonacci lifts are dense in SU(2). Its exact entries are algebraic numbers. For the finite bit implementation, the real and imaginary coordinates are supplied by minimal polynomials over $\mathbb{Q}$ and rational isolating intervals with opposite endpoint signs and exactly one real root, as required by the finite-precision SK theorem. For the SK recursion one additionally supplies a fixed finite base-net certificate $({\mathcal{N}}_0,{\eta }_0,L_0)$ with $0<{\eta }_0<{64}^{-2}$, as defined in Inverse-Closed Dense SU(2) Gate Set. The certificate is input data: the compactness result Compactness Gives a Finite SU(2) Base Net proves existence but computes neither the list nor $L_0$.

The diagram uses the positive Artin crossing convention: time runs downward, and the continuous left-input strand passes over the broken right-input strand. Its word matrix is the representation convention in Braid action on a fixed-charge fusion space; no proof depends on the drawing.