Recurrence: An Evolving Research-State Copilot for Mathematics

Much of recent AI-for-math frames theorem proving as a single proof obligation: given a statement, generate, check, and revise a proof. In mathematical research, however, progress often depends on the surrounding state that comes before and between proof attempts: fixing the right statement, testing examples and counterexamples, remembering failed routes, extracting reusable lemmas, and deciding what should be attacked next. We present Recurrence, a self-evolving copilot for this layer of mathematical work. Recurrence adds a pre-proof layer to a proof-and-verification pipeline: it locks the target statement, separates exploratory mapping from direct attack, keeps proof-side and disproof-side routes visible, labels claims through adversarial verification, and stores memory items that guide later runs. Its main output is not only a proof attempt, but a reusable record of the proof attempt, counterexample search, verifier objections, failed routes, remaining bottlenecks, and the recommended next target. By “self-evolving” we mean that the research state, prompts, and memory evolve across runs; no model weights are updated. We report pilot evidence from topology campaigns and a cluster-benchmark protocol instantiated on a 22-target Gompf problem list, showing how such artifacts preserve obstructions, avoid false solution claims, and produce human-checkable next steps. Code: https://github.com/Recurrence-Poincare/Recurrence-Agent.