Basins with tentacles

The paper numerically shows that, for the nearest-neighbor Kuramoto ring, the relative basin sizes of q–twisted states scale like e^{-k q^2} and not like e^{-c|q|} (global sampling vs. local hypercube estimates) and explains the discrepancy via “tentacle”-like basin geometry. These claims are documented (model (1), stability window |q|<n/4; global sampling strongly favoring Gaussian dependence; hypercube-based estimates missing essentially all volume) and are consistent across the paper’s figures and discussion . The model’s solution gives an explicit, rigorous expression B_n(q)=2π f_{n,π}(2π q) (sum-of-uniforms/Irwin–Hall), yielding a local-CLT Gaussian with variance ∝ n and showing why a pure e^{-c|q|} law cannot hold. It also quantifies why hypercube methods capture an exponentially vanishing fraction of the true basin volume. Thus the paper’s empirical findings are explained and supported by the model’s derivation; they are not in logical conflict (the paper never asserts an exact Gaussian law across the entire admissible q-window).