Murmurations in elliptic curves were discovered in 2022-2023: oscillatory patterns in average Frobenius traces that vary with analytic rank and conductor. The patterns are empirical—no complete theoretical explanation exists. They look like statistical noise until you sort by rank, at which point coherent waves appear.
Bieri, Costa, Deines, Lee, Lowry-Duda, Oliver, Qi, and Veenstra train one-dimensional convolutional neural networks on Frobenius trace sequences and find they predict analytic rank with high accuracy. This alone is notable but not surprising—neural networks find patterns.
The surprise is in the saliency maps. When the authors examine what the network learns to detect, the saliency curves reveal the network is detecting murmurations. It independently rediscovered the same oscillatory structure that human mathematicians found through different methods. Furthermore, the paper connects murmurations to Mestre-Nagao sums—a 1990s tool for rank prediction—showing the relationship between them varies with conductor and predicted rank.
Three independently discovered phenomena—murmurations (2022), neural network saliency patterns (this paper), and Mestre-Nagao sums (1990s)—turn out to be shadows of the same underlying structure. The network provides evidence for a theorem that hasn't been proved yet: whatever generates murmurations carries enough information to determine rank, and it has been implicitly used by Mestre-Nagao sums for three decades without anyone recognizing the connection.
When a machine learning model trained on task A independently discovers pattern B, it suggests A and B share latent structure. The network doesn't prove the theorem, but it demonstrates the theorem's conclusion.