Every anyon theory carries modular data (an S-matrix and a T-matrix) that must satisfy hard algebraic consistency conditions: S is symmetric and unitary, S² = (charge conjugation) so S² = I here, and the Verlinde formula reproduces the fusion rules (for the non-trivial anyon, τ × τ = 1 + τ). The lights on the left show these all pass for the Fibonacci theory, and the S-matrix readout gives dτ = φ and chiral central charge c = 14/5.
The headline control is the bar chart: the smallest distance a braid word (up to a fixed length) can get to the single-qubit T-gate. Fibonacci braiding gets arbitrarily close: it is dense in the unitary group, hence universal. Ising braiding is pinned far away; it generates only the Clifford group, so the T-gate is provably out of reach by braiding alone.
The point: the Ising theory has equally consistent modular data, a perfectly valid S and T. Consistency does not buy you universality. φ is load-bearing in the universality gap (the Fibonacci anyon's quantum dimension d = φ is exactly what makes its braid group dense), not in the bare consistency of the modular data. Everything here is computed live in your browser via WebAssembly.