SU(2) Fibonacci Solovay–Kitaev compiler

Inputs: a positive binary-rational tolerance $0<\epsilon \le {\eta }_0$; the fixed inverse-closed Fibonacci alphabet $G_{\mathrm{F}\mathrm{i}\mathrm{b}}$; and a fixed certified base-net certificate $({\mathcal{N}}_0,{\eta }_0,L_0)$ with $0<{\eta }_0<{64}^{-2}$. Select exactly one target interface:

1. Ideal exact-real interface: supply an exact-real handle for $S\in SU(2)$ with exact fixed-dimensional matrix operations and exact comparisons. Invoke the ideal recursive procedure Solovay–Kitaev Unitary Compiler for SU(2) on $S$ and $\epsilon$.
2. Finite Turing interface: supply a polynomial-time Cauchy name for $S\in SU(2)$ and the minimal-polynomial and rational isolating-interval descriptions of every real and imaginary gate coordinate required by the finite-precision theorem. Run the deterministic Turing implementation established by Finite-Precision Bit Complexity of the SU(2) Solovay–Kitaev Compiler on that name, alphabet, certificate, and tolerance. This branch uses its certified interval decisions and does not invoke exact-real comparisons.

In either branch, return the resulting word in the labels $z{B}_1,z{B}_2,(z{B}_1{)}^{-1},(z{B}_2{)}^{-1}$, its expanded length, and its Artin exponent sum $k$. The base-net list is supplied certificate data; this algorithm does not compute it from density. A Cauchy name alone is not an exact-real handle, and the ideal branch makes no finite bit-time claim.

Output: a braid word $b$ and ledger scalar $(R_1{z}^{-1}{)}^k$ such that, when the algorithm's hypotheses hold, the normalized SU(2) matrix of $b$ is within the requested tolerance of $S$. Its physical matrix carries the recorded scalar from Braid phase reference and correction ledger.