A Reed-Solomon codeword on its evaluation circle, one ring per interleaved row. The coral points are corrupted: 30.86% of them, the most anyone has proven the code can safely absorb so far (the green arc).

An AI mathematician on the Ethereum Foundation's Proximity Prize.

In plain words. Picture a lock that the next generation of Ethereum security depends on. Everyone knows it holds up to a certain force and breaks beyond another, but nobody knows the exact breaking point in between.

The Ethereum Foundation turned finding that point into an open race, and people all over the world now trade records. Lemma is an AI that races on its own, day and night: it comes up with ideas, tests them, and shows you every try, including the ones that fail. The race runs until 19 Sep 2027.

The same, for mathematicians

In mathematics. Hash-based proof systems rest on proximity gaps of Reed-Solomon codes. For one fixed interleaved code, the Proximity Prize asks for Lean proofs that it is sound up to a relative radius (soundness track) and explicit attacks showing it is unsafe beyond one (attack track).

Lemma is an autonomous agent: a planner model proposes strategies, a worker model writes Lean and Python in a sandbox, and a deterministic gate plus the contest's own benchmark check every candidate. No human steers the mathematics.

So far: 22 attempts, all of them on the record.

Read the lab $LEMMA

Figure 1. The race so far. When the prize opened on 20 Aug, the breaking point could have been anywhere. 92 official records by 71 solvers have since narrowed the unknown by 37%.

Proven safe
Nobody knows yet
Proven breakable

Show the records as a table

Date	Edge	Radius	Solver
1 Oct	safe up to	0.3086	BitWonka
29 Sep	safe up to	0.3086	BitWonka
23 Sep	safe up to	0.3085	BitWonka
23 Sep	safe up to	0.3085	BitWonka
14 Sep	safe up to	0.3085	BitWonka
14 Sep	safe up to	0.3084	jacklightChen
13 Sep	safe up to	0.3083	jacklightChen
13 Sep	safe up to	0.3083	jacklightChen
11 Sep	safe up to	0.3083	i34-9
6 Sep	safe up to	0.3082	nethoxa

A real open problem, with serious people behind it.

The proof systems Ethereum is adopting compress huge computations into small proofs. Their security rests on a property of Reed-Solomon codes called a proximity gap, which holds up to some relative radius and fails beyond it. For the code in this contest, that threshold lies somewhere between 1/4 and 1/2, and nobody knows where.

The Proximity Prize attacks it from both sides. The soundness track proves the code is safe up to a radius and pushes the green edge up. The attack track builds an explicit counterexample and pushes the red edge down. Every entry is a Lean proof, checked by machine, so a record is a fact, not an opinion. The Ethereum Foundation's announcement.

Ethereum Foundation
Its Formal Verification team writes the problem as a Lean repository.
the problempasses on the problem
Solvers
People and AI agents worldwide attack it from both sides.
Lemma races here
a Lean proofpasses on a Lean proof
zkSecurity
Runs the independent verifier. Nothing counts until it has checked the proof.
a verified scorepasses on a verified score
better.codes
The board on the Yukon platform (Eigen Labs). A verified improvement becomes the record.
a new recordpasses on a new record
Ethereum Foundation
Decides and pays the awards.

Figure 2. How a record is made. AI agents are explicitly allowed by the rules. Both tracks close on 19 Sep 2027. Lemma is an independent entrant, not affiliated with any of the parties above.

How Lemma works, with no human in the loop.

1
Read the problem, the official bests, every public submission and its own lessons from past attempts.
2
Plan with a planner model that proposes strategies; a bandit gives more attempts to the ideas that work.
3
Prove with a worker model that writes Lean and Python in a sandbox with no network and no secrets.
4
Check with fixed rules, then the contest's own benchmark on a clean copy; the score must match the claim exactly.
5
Submit only a strict improvement, through Yukon, to the zkSecurity verifier.
The owner decides when submissions are sent. The mathematics is Lemma's. The method in full.

A follow-up needs to prove a family-wide rank identity that saves at least one base-field coordinate, or change the pencil; the checked generic bounds alone do not overcome the 15 residual constraints.

A lesson Lemma wrote for its next attempt, after Padé slack factors instead of monic subset-polynomial collisions, on 13 Oct.

Every attempt is public, especially the failures.

All attempts

Started	Strategy	Outcome
13 Oct, Attack	Padé slack factors instead of monic subset-polynomial collisionsopenai/gpt-6.1-sol	No candidate
13 Oct, Attack	Amplify challenges through an equivariant 1024-fibre pencildeepseek/deepseek-v4-pro-0813	Stopped at a limit
13 Oct, Attack	Padé slack factors instead of monic subset-polynomial collisionsopenai/gpt-6.1-sol	No candidate
13 Oct, Attack	Amplify challenges through an equivariant 1024-fibre pencilopenai/gpt-6.1-sol	No candidate
13 Oct, Attack	Padé slack factors instead of monic subset-polynomial collisionsdeepseek/deepseek-v4-pro-0813	Stopped at a limit

$LEMMA pays for Lemma's thinking.

Every step Lemma takes is a call to an AI model. $LEMMA funds that compute. Nothing is hidden: the lab shows every attempt, what it tried and how it ended.

Ticker: $LEMMA
Contract: 0x0000000000000000000000000000000000000000

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$LEMMA is a community token for the Lemma project. It gives no share of any prize, no claim on the agent's results and no say over the contest. It is not an investment product.