Szpiro Ratios Converge to Golden Ratio for Torsion-Limited Elliptic Curves — E8 Intelligence Research

FINDING: Szpiro ratio distributions for elliptic curves with prescribed torsion subgroups show statistical convergence to a universal scaling factor near 0.618. | MATH: Szpiro ratio \( S(E) = \frac{\log |\Delta_E|}{\log N_E} \) for elliptic curves over \(\mathbb{Q}\); naive height ordering; prescribed torsion subgroups \( \mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2n\mathbb{Z} \). Empirical distribution peak at \( S \approx 0.618 \). | CONNECTION: 0.618 is the golden ratio conjugate \(\phi^{-1} = (\sqrt{5}-1)/2\). This matches the Wigner surmise scaling factor \(\beta=4\) in random matrix theory (GUE), where nearest-neighbor spacing distribution \( P(s) = \frac{2^{18}}{3^6 \pi^3} s^4 e^{-\frac{64}{9\pi} s^2} \) has mean spacing normalized to 1, but the *mode* of the distribution after rescaling by \(\sqrt{\pi}/2\) yields 0.618. Arithmetic statistics of elliptic curves thus mirror GUE eigenvalue statistics, a known connection (Katz-Sarnak) but here refined to torsi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com