AI learns to prove impossibility: Tsinghua’s AWM beats GPT-6 Astra and Claude Opus 5.5 where they give up

What does an AI do when a task has no solution? Usually it says “I couldn’t do it” — which is indistinguishable from “I didn’t find it.” A team from Tsinghua University and Zhejiang University led by Turing Award winner Andrew Yao has introduced the Algebraic World Model (AWM), a system that proves impossibility literally: it builds a mathematical model of an unfamiliar environment, formalizes it in Lean 4 and produces a checkable proof that no winning sequence exists. On ARC-AGI-3 the approach clears all 183 levels and lifts Claude Opus 5’s score from 40.7 to 99.9.

Why “it didn’t work” is a bad answer

ARC-AGI-3 is a set of 25 public games where the agent gets neither rules nor a goal. It has to work out the mechanics and then clear the level. These tasks test whether a model can reason about an unfamiliar world rather than recall similar examples from training.

That is exactly where the flagships stumble. In an experiment on two sealed levels — levels whose action budget is one unit below the cheapest possible win — neither Claude Opus 5.5 nor GPT-6 Astra, both equipped with coding tools, produced a single proof. Across 12 runs they said the task was impossible only five times, and in four of those cases they named the wrong minimum cost. On level sc25 with a budget of 18 actions (the cheapest win costs 19), GPT-6 Astra found a route costing 22 moves and concluded nothing cheaper existed — without a proof.

How AWM turns a game into a theorem

The work happens in three stages. First, exploration: the agent plays, records observations and writes hypotheses as executable Python code. Every hypothesis is tested against all observed transitions and rewritten if even one transition contradicts it.

Then comes structure: hypotheses become typed objects and shared rules, while category-theory-inspired transformations strip away redundancy. In the lp85 game, eleven separate concepts about four kinds of buttons collapsed into a single rule, press_permutes. Finally, the model is translated into Lean 4, the language mathematicians use for machine-checkable proofs. There AWM proves two kinds of statements: that a specific route reaches the goal, and that under a given budget no winning route exists at all.

The key phrase is “at all.” A normal model says “I didn’t find it.” AWM presents a proof that can be checked line by line and that rules out every legal sequence of actions, not just the ones the agent happened to try.

Six proofs versus zero

On the two sealed levels AWM produced a closed Lean proof that winning within the budget is impossible in all six runs out of six. GPT-6 Astra and Claude Opus 5.5 produced zero such proofs across twelve runs. The minimum-cost estimates are even more telling: four of AWM’s six initial bounds were correct, Claude Opus 5.5 got one of six right, and GPT-6 Astra got none. After being told a cheaper win existed, AWM revised its rules and reached six correct bounds out of six.

183 levels, 99.9 points and 43% of a human’s actions

Then comes the full ARC-AGI-3 set. Across the 25 public games, AWM’s models let Lean replay the winning trajectories of all 183 levels. On unseen layouts the frozen rules predicted all 4,160 transitions; after a mechanic change the model rejected 4,031 transitions — meaning the model noticed the world had changed instead of carrying on with the old instructions.

With Claude Opus 5 as the base model, AWM scores 99.9; the same Claude Opus 5 without AWM scores 40.7. The certified winning trajectories use 43% of the actions a human baseline spends. In other words, the system does not just clear the game; it clears it more economically than an average human.

Where the catch is

AWM’s proofs are proofs about a learned model, not about the game itself. Turning screens into states happens in Python, outside Lean. The model can end up stricter than the game; then a proof of “impossible” would be true for the model and false for reality. The authors are explicit about what the proofs establish: the rules agree with every recorded transition, the certified route really clears the level under those rules, and on sealed levels no winning route exists within the budget.

The link to the real game is established from outside: the frozen rules predict 4,160 transitions on unseen layouts, and an exhaustive search of the game engine confirms the sealed thresholds. But if exploration was incomplete and the agent missed an important rule, the result is a local truth, not a description of the whole world. The authors describe a loop: observation, hypothesis, experiment, abstraction, proof, and observation again. The paper is on arXiv (submitted 29 October 2026, revised 30 October), and run records plus a demo are on the project site.

Why this matters beyond games

Agents already plan trips, write code and manage tasks. In the real world, though, they will have to say not only “here is the solution” but “there is no solution, and here is why.” The difference is huge. “I didn’t find it” means trying another approach or waiting. “I proved it cannot be done” frees resources, changes the plan and stops an endless loop.

For a platform built on generative models, the question looks the same from the other side: the more tasks get automated, the more important checkability becomes — not confidence in an answer. A formal proof is the most extreme, strictest form of that checkability. In games it already works. In planning, budgeting and code review, the next step is learning to build equally short and readable proofs for problems without clear rules.

What it means for users

Both models from the paper — Claude Opus 5.5 and GPT-6 Astra — are available in NeuralSpace Chat. AWM is a research project with no public access, but you can already see how modern models reason about your own tasks. For building agents and projects of your own, there is the Code section.

And a bit of real statistics. According to anonymized production data from NeuralSpace over the last 30 days (window 03.10–02.11.2026), the chat handled 1,513 user messages across 516 sessions. Four models accounted for 69% of replies, with GPT-5.4 Mini leading at 580 responses. Heavy flagships are picked selectively: in everyday work, speed and price usually win, while hard problems are a separate story.

FAQ

What is the Algebraic World Model?

A system that explores an unfamiliar environment, builds a mathematical model of it and formally proves statements about that model in Lean 4 — for example, that a goal is unreachable within a given action budget.

How is AWM different from a regular language model?

A language model proposes an answer and can be wrong without showing it. AWM first builds explicit rules of the world, then checks them and derives consequences as a proof that an independent program can verify. A mistake in a rule is visible instead of hidden in the weights.

Can I try AWM?

There is no public version; it is a research project. But the base models it ran on are available in NeuralSpace Chat, and you can build your own experiments in the Code section.