Murmurations of elliptic curves

Stage M1 — replicating He–Lee–Oliver–Pozdnyakov (arXiv:2204.10140), Figure 1. The mean of ap over rank-classed elliptic curves in a conductor window oscillates with p; even/odd rank oscillate in antiphase. ap is computed from scratch (good primes) and Atkin–Lehner-converted (bad primes); rank and conductor are oracle (Cremona ecdata). Renders emitted JSON only.

HLOP Figure 1 (top) — conductor , ranks 0 & 1

f₀ (rank 0) f₁ (rank 1) — antiphase

HLOP Figure 1 (bottom) — conductor , ranks 0 & 2

f₀ (rank 0) f₂ (rank 2) — same parity, in phase

Scale collapse in y = p/N, per rank (two controls)

Disjoint windows [2500,5000] vs [5000,10000] are consistent with p/N scaling (ratio = RMS distance / sampling-noise floor F; < 3). Two controls: the reversal null (within-curve scramble) measures the test's power against a wrong scaling — it turned out limited (ratio < 3, not rejected) at these family sizes; the parity null pairs the opposite-parity rank and tests non-degeneracy (differs in rank AND range, so it is NOT a scaling null). Rank 2 is low power. F assumes independent ap (anticonservative — the murmuration IS correlation); ratios are observed empirics, not proof.

Family counts & provenance