Learning RSI: mathematics as a laboratory for recursive improvement (The Sequence Knowledge #968)

By TheSequence · Tue, 10 Nov 2026 12:05:44 +0000.

Issue #968 of The Sequence Knowledge argues that, given a hard theorem and enough time, a language model can produce a remarkable amount of mathematics. The real challenge is deciding which of that material is worth keeping. An elegant paragraph can hide an invalid inference, and a correct final answer can emerge from a broken argument. Training on either of these errors teaches the next model to repeat them even more convincingly.

According to the text, mathematics is an especially useful laboratory for recursive improvement because proposals can be independently verified. A generator (the prover) proposes candidates, and a verifier determines whether they establish a precise statement. Successful attempts become new training material. The issue examines three systems that explore variants of this scheme: AlphaProof, Goedel-Prover-V2 and DeepSeekMath-V2. Their differences show where recursive improvement gains traction: feedback must remain informative as the system generating candidates grows stronger.

The email summary previews the topics covered in the full edition: how the Lean kernel creates a useful asymmetry between finding a proof and checking it; how AlphaProof turns exploration into training, including test-time reinforcement learning; why useful theorems must also be discoverable; and what changes when verification itself is learned. It closes with the limit a verifier cannot cross, namely whether the stated theorem is the one actually intended, and with the measurements that would demonstrate genuine recursive benefit.

Note: the body of the received email is truncated (it ends at "Proof checking changes the learning problem..."), so this summary is limited to the introduction and table of contents; technical details on each system are in the web version of the article.