Empirical short-interval average (per prime) vs the density M₂(y)
empirical (per prime P)
empirical (smoothed)
M₂(y) density (formula)
teeth at n²/4
Stage M3 — Zubrilina (arXiv:2310.07681), Theorem 1. The root-number-weighted average of af(P) over weight-2 newforms of square-free level in a short interval, as a function of y = P/X, converges to the density M₂(y). The empirical curve is assembled entirely from class numbers — the Eichler–Selberg / Skoruppa–Zagier trace formula evaluated per level, normalized by dim S₂new — with zero external eigenvalue data. Its derivative discontinuities (teeth) sit at y = n²/4 = {¼, 1, 9⁄4}. Renders emitted JSON only.