NVIDIA Open-Sources IMO Gold-Level Math Reasoning System, Exposing Compute and Verification Gaps

The model wrote proofs in natural language only, without calling a formal prover such as Lean and without external tools or web retrieval, the report said. The release is not just a stronger math model. Nemotron 3 Ultra is the base, while two expert models, large-scale proof search, model verification and multi-round rewriting form the chain that pushed the score to the gold line.

NVIDIA did not repeatedly sample the same Nemotron 3 Ultra. It first trained two math experts with visibly different behavior. In the Nemotron design, post-training directly serves later search. SFT data include not only complete proofs but also many traces of proof modification, verification and re-verification. A model trained this way learns not only to generate a proof from a problem but also to inspect a half-written or locally incorrect proof, locate the problem and continue repairing it along the existing route.

The RL expert addresses another layer: how to change the probability that proof routes are sampled. Proof space for IMO-level problems is sparse. A model may already have the local abilities needed for a class of problems, but the correct combination occupies only a small part of the generation distribution. Reinforcement learning reweights the distribution based on successful and failed trajectories. Ideas that close a proof receive higher weight; choices that repeatedly lead into dead ends are suppressed. It does not create new mathematical knowledge, but rearranges the frequency with which existing abilities appear.

The general, SFT and RL checkpoints therefore have three different problem-solving preferences. That matters because search depends on effective sample size, not just the number of generated answers. If one model produces 200 proofs that mostly circle a few constructions, those 200 texts are not 200 independent routes. Adding checkpoints trained differently changes the sampling distribution and lets compute enter another part of proof space. In NVIDIA's experiments, increasing sampling from the same RL model quickly slowed returns, while adding the SFT expert increased the number of problems covered even with a similar generation budget.

Nemotron generates 384 proofs per problem in the first round, but those proofs are not the endpoint. They enter a continuously updated proof pool. After verification, each proof falls into one of three cases: it passes directly; its overall direction is usable but has defects; or its route has low value. The system does not simply delete the latter two categories. It keeps higher-scoring proofs and sends the problems identified by the verifier back to the model for further modification.

This changes the nature of inference. Ordinary repeated sampling starts over from the beginning each time, with almost no memory between generations. Nemotron's proof pool preserves previous computation. Where a route has reached, what went wrong and which parts remain usable all become inputs for the next search round. The verifier's corrections act as a direction signal. Natural-language proofs do not have a continuous differentiable objective function, so the system cannot directly calculate a gradient-like next step. Correction comments approximate that role by telling the model where the current proof still falls short of an acceptable answer.

Multi-round rewriting is therefore not polishing the same answer. Each round produces multiple candidates that re-enter the global proof pool and compete with earlier routes. If a proof continues to receive high evaluations, compute keeps flowing along that path. If several rounds fail to fix a critical gap, it gradually loses the chance to expand. The system thus has both search width, from hundreds of first-round proofs, and search depth, from later refinement.

The verifier also creates a limitation. Unlike Go, natural-language mathematics does not have a cheap rule system for judging legality or a clear endgame. A proof may be valid for dozens of steps and then use a non-existent symmetry, or a step may be written loosely while the overall mathematical idea still holds. The verifier therefore not only scores answers but also influences where computation goes. Its errors can affect subsequent search paths, and the larger the search, the more visible that influence becomes.

NVIDIA set a high acceptance threshold. Two experts repeatedly check the same proof, and only when all judgments pass is the candidate accepted, a standard described as 16/16 unanimous. This reduces the chance of admitting false proofs but also rejects many correct proofs. The deeper problem is that verification judgments are not independent. The SFT, RL and general checkpoints share the same Nemotron 3 Ultra base, with similar knowledge structures, concept representations and reasoning habits. When an error comes from random oversight, multiple checks help. When it comes from a shared blind spot, more checks help much less.

The report cites a false proof involving a permutation symmetry. A counterexample could be constructed explicitly, but the three checkpoints repeatedly judged it and none identified the key flaw. On Problem 6, the system continued to spend large amounts of computation and produced a new proof. Internal verification still did not pass it, but human review found that a substantial part was correct and deserved partial credit. The verifier thus shows two kinds of bias: some false proofs pass with high confidence because of shared blind spots, while some valuable proofs are suppressed by an overly conservative threshold.

NVIDIA's release includes the two expert checkpoints, SFT and RL data, inference code, the RL recipe, submitted proofs and 200 Nemotron-IMO-Bench problems. External teams can vary checkpoint combinations, sampling budgets, verifier thresholds and refinement depth, then observe how scores change. But reproducing the 30/42 result remains different from reading the recipe. A 550B-class model, terabyte-level VRAM and thousands of GB200 GPU hours still form a high hardware barrier. NVIDIA's official documentation also states that part of the full post-training chain for the general Nemotron 3 Ultra model, including some intermediate teacher and MOPD checkpoints, has not been opened.

Two days after the release, on September 11, 25 Fields medalists including Terence Tao, Peter Scholze and Maryna Viazovska jointly criticized the accelerating pace of AI mathematics announcements, saying proof checking, method review and reproduction often do not have enough time. Leiphone reported that even with the full recipe, 8 B200 GPUs and 1,464 hours of long inference remain a hard wall for most university laboratories. The report described the situation as code equality already achieved while compute sovereignty remains concentrated among a few large companies.