What does this equation mean?

The first verdict is a genuine win for the theory. Every exponent measured is super-linear — $\zeta$ > 1 at seven of the eight points — exactly where the long-range bound says a finite-velocity linear cone is not guaranteed and faster-than-linear growth is required instead, and the propagation outright violates the nearest-neighbour bound $v_{\mathrm{LR}}$ = 12e$J_{\max}$ once $\alpha$ drops below 1 [ 4 ] . Read against a naive short-range picture, the ion chain is unambiguously non-local. The second verdict tempers the first: the three competing long-range thresholds theory offers — a worst-case bound, a typical-state bound, and a free-particle bound — all sit above $\alpha$ = 2 , while Richerme’s scan stops at $\alpha$ = 1.19 , so the data occupy a regime where all three theories predict the same qualitative behaviour and the experiment cannot say which one is actually being tested. Discriminating between them would need a scan crossing a threshold, which this dataset was never built to provide. The third verdict resists tidying altogether: the XY measurement at $\alpha$ = 1.19 gives $\zeta$ = 1.67 $\pm$ 0.08 , growing super-linearly even though a naive count of the interaction’s range would call it short enough to behave, and no published theory accounts for it. The confrontation names that gap rather than explaining it away.