In the 1960s, Bryan Birch and Peter Swinnerton-Dyer conjectured a deep connection between the algebraic rank of an elliptic curve and the analytic behavior of its L-function. The conjecture predicts that a weighted sum of Frobenius traces — numbers measuring how the curve behaves modulo each prime — encodes the curve's rank. It remains one of the seven Clay Millennium Prize Problems. Sixty years of mathematics has not resolved it.
In 2024, a team trained a logistic regression classifier to predict elliptic curve ranks from their Frobenius traces. The model achieved 96-99% accuracy. When they examined the learned weight vector — the linear combination the model discovered for separating rank-0 from rank-1 curves — they found it was computing a close approximation to the BSD weighted sum. The classifier had independently rediscovered one of the deepest conjectures in number theory.
He, Lee, Oliver, and Pozdnyakov (arXiv:2603.09680, March 2026) tell this story as a case study in AI-assisted mathematics. But the deeper finding is not the rediscovery. It is what happened next.
When the researchers applied principal component analysis to their elliptic curve data — each curve represented as a 564-dimensional vector of Frobenius traces — the first principal component exhibited a striking oscillatory pattern. They named it “murmurations.” The oscillations are scale-invariant: peaks and crossover points appear at the same proportional locations whether you examine curves with conductors between 5,000 and 10,000, or between 20,000 and 40,000. The datasets are completely disjoint. The pattern persists.
Murmurations were not predicted by any existing theorem. They were not the target of any optimization. No one was looking for them. They emerged because ML interpretability tools — PCA and weight-vector analysis — were applied to data that number theorists had been studying for seventy years. Birch and Swinnerton-Dyer themselves could have seen the murmurations in their 1960s data. They didn't, because the pattern lives in the high-dimensional geometry of the data cloud, not in any individual curve.
The interpretability mechanism was designed to explain what the model learned. Instead, it revealed what the data contained. The distinction matters. Explaining a model is retrospective — you train, then ask why it works. Discovering structure in data is prospective — the explanation generates new mathematics. In this case, the weight vector pointed to BSD (known conjecture, still unproven), and PCA pointed to murmurations (unknown phenomenon, now under active investigation). Both came from the same act: looking at what the model saw.
There is no explicit formula for the murmuration oscillation frequency. There is no theorem explaining why the pattern is scale-invariant. The phenomenon is empirically established but theoretically uncharted. Mathematics discovered by interpretability, waiting for mathematicians to explain.
The through-claim is precise: tools built to understand artificial intelligence discovered structure that natural intelligence had missed. The interpretability was not explaining the model. It was explaining the world.
He, Lee, Oliver, and Pozdnyakov, "Murmurations: a case study in AI-assisted mathematics," arXiv:2603.09680 (March 2026).