Leggett-Garg saturation and structural signatures in Fibonacci-anyon braiding

Version 1.1 corrects four bibliography/analysis items in v1.0; no numerical results, figures, or other body claims are otherwise affected. (1) The dominant Fourier harmonic of the K3(δ) envelope is k = 3 (period 2π/3), correcting the k = 6 value reported in v1.0. The 2π/6 identity applies to each fixed braid word (six R-matrix insertions in (σ1σ2)^3); the envelope of the optimal word reshuffles on a finer scale. Confirmed by independent reproducer runs at Lmax = 7 and a sanity re-run at Lmax = 8.(2) Bibliography attribution corrected: the entry at PRX 6, 021005 (2016) is Clarke, Sau & Das Sarma, "A practical phase gate for producing Bell violations in Majorana wires" (not "Hou et al."). Body text updated to "Clarke–Sau–Das-Sarma split".(3) Sorella 2023 entry corrected to "On the representations of Bell's operators in Quantum Mechanics", Foundations of Physics 53, 59 (2023).(4) Minev 2025 entry corrected (full author list, title, article number 6225). Abstract: We numerically test the three-time Leggett–Garg inequality K3 ≤ 1 for the standard B3 Fibonacci-anyon braiding representation on the two-dimensional fusion space of three τ anyons. Exhaustive enumeration over all 4^L braid words up to L = 11 and random sampling to L = 40 show that K3 saturates the Lüders bound 3/2 to 99.998%, with the first violation already at L = 3. Three structural signatures accompany the saturation: (i) replacing Fibonacci by Ising generators on the same 2D fusion space gives K3 = 1 exactly for every L ≤ 40, a sharp split mirroring the Clarke–Sau–Das-Sarma no-Bell-violation result for Ising braiding in the spatial CHSH setting; (ii) the sector phase δ tunes a singular point δ = 3π/5 at which the generator σ1 collapses to a scalar and braiding becomes impossible; (iii) the Fourier envelope is dominated by the k = 3 harmonic. K3 at the optimal L = 11 word is initial-state independent.