Near marginal stability, finite systems fluctuate anomalously — not following the standard N^{-1/2} central limit theorem scaling, not the mean-field N^{-1}. These anomalous fluctuations were known. What was not known was how stubbornly they persist as the system grows.
The critical window shrinks as N^{-1/5}.
In long-range interacting Hamiltonian systems near marginal stability, the region where fluctuations are anomalous — non-Gaussian, violating standard statistical expectations — contracts at the fifth root of the inverse system size. This is extraordinarily slow. At N = 10⁵ particles, the anomalous window has shrunk by a factor of only 10. At N = 10¹⁰, by only 100. The law of large numbers is technically winning, but it is winning slowly enough that for any experimentally accessible system size, the anomalous behavior dominates.
The mechanism runs through the coupling between marginal stability and collective fluctuations. Near the stability boundary, small perturbations produce responses that are neither independent (which would give N^{-1/2}) nor perfectly correlated (which would give N^{-1}). The fifth-root exponent captures the intermediate regime where correlations decay slowly enough to resist statistical normalization but fast enough to avoid complete synchrony.
Outside this window, fluctuations are Gaussian. The transition from anomalous to normal is not sharp — it is governed by this slowly-shrinking envelope. The window is a soft boundary that retreats but refuses to collapse.
For systems living near marginal stability — neural networks at the edge of chaos, financial markets near criticality, ecosystems near tipping points — the practical implication is that anomalous fluctuations are not a finite-size artifact. They are the relevant physics at any achievable scale. The large-N limit exists. It is just unreachable.