Mean residual error compiling random reachable targets on the D-dimensional fusion qudit, at a matched search budget — lower is better. The co-adaptive beam (grow a prefix and a suffix together, each scored against the other) gets meet-in-the-middle's reach and beam pruning, so it beats both one-directional exhaustive search and static meet-in-the-middle. Solovay–Kitaev does not apply at D > 2. (Full D = 2…8 sweep: package benchmark bench_coadaptive_synth.)

The dual-mode core compiles a single quantum intent (a Hadamard, a T-gate, a φ-pseudo-Haar twirl) to two backends at once: the familiar circuit (gate) mode, and the anyonic (braiding) mode that realizes the same unitary by braiding Fibonacci anyons on a shared weave. Both reach the target at fidelity ≈ 1.0; the only difference is a tiny residual braiding error from compiling the target into a finite braid word.

The top chart zooms the y-axis right up against 1.0 so the two modes are distinguishable; the second chart plots 1 − fidelity on a log axis so the braid error, invisible at full scale, becomes legible, annotated with each target's braid length and the cross-mode agreement.

The third chart is the entangling weave. A two-qubit gate can't come from single-anyon moves; it needs a genuine braid of six τ-anyons, searched live. As the allowed braid grows longer, the realized gate's distance to the CNOT local class collapses (here ~1.9 → 0.19 once the braid reaches length 7), while the output stays certified entangling throughout: Schmidt rank ≥ 2 and positive PPT-negativity on a separable input. Entanglement straight out of braiding, no two-qubit gate ever applied.

Where φ is load-bearing: only in the anyonic mode. The braiding backend lives on the Fibonacci anyon theory whose total quantum dimension is exactly D = √(1 + φ²) ≈ 1.902. That golden-ratio fact is what makes braiding alone universal. In the circuit mode φ plays no special role; it is one gate set among many. Everything here is computed live in your browser from phi-quantum-core-cpp via WebAssembly.