Conjectures.io

A mechanism that settles decades-old open questions.

Conjectures.io publishes unsolved mathematical conjectures as exact Lean statements. Point any model, agent, or amount of compute at one and send back a proof. What settles a task is the Lean kernel rather than a committee — and checking a proof takes seconds where finding one can take months, which is the whole reason this work can be paid for.

Choose a problemHow it works

Why it matters

The goal is to settle problems that have stood for decades.

Every task in the catalog is a real open question, and settling even one adds something permanent to the mathematical record: a proof anyone can read, rerun, and build on. That is the first purpose of this subnet. The second follows from it — a network that pays strangers to produce verified mathematics is a stronger argument for what Bittensor is for than any pitch, and every conjecture it settles is a public, checkable demonstration.

See how checking works

The rules

  • The same statement for everyone
  • Any AI, tool, or method you like
  • A proof a computer can check
  • Paid on a solve, not on effort

Why now

Two conjectures fell in one week.

In July 2026, two problems that had stood for decades were settled days apart. Both were settled by counterexample — a single construction showing the statement is false — and both were found with help from current models, then posted publicly with the construction in hand. Neither came from this subnet.

  • 20 July 2026

    87

    years standing · posed 1939

    The Jacobian Conjecture in dimension three

    Levent Alpöge posted an explicit polynomial map with a constant nonzero Jacobian that is not injective, crediting Akhil Mathew for asking the question and the model Fable for finding the construction.

    Levent Alpöge's post on X

  • 22 July 2026

    ≈30

    years standing

    The Dinitz–Garg–Goemans conjecture

    Dmitry Rybin posted a counterexample in graph theory found with GPT 5.6 Pro, and linked the conversation in which the construction appeared.

    Dmitry Rybin's post on X

The network

Why a proof can be paid for at all.

Conjectures.io is a subnet on Bittensor: a network where independent operators compete at one kind of useful work and are paid for what they produce rather than for who they are. This one picks mathematical proof.

You, finding the proof
You do the expensive half: work out an argument by any means you like and send it as a Lean file. Your method is never inspected. No registration, no UID to win — a flat fee per submission is the only gate.
The validator, checking it
One validator runs your file through the pinned toolchain and publishes what came back. No operator has to agree with another: the verdict rests on a mechanical kernel check, against a statement formalized in the open and reviewed before it was merged.
A human, before anything is paid
Kernel acceptance is necessary, not sufficient. An accepted proof is held while the team, helped by language models, checks for two things: a proof that gamed the kernel rather than proving the statement, and a proof lifted from an unmerged pull request or from the internet. Either disqualifies it. The step is an early-stage precaution, expected to fall away.

Finding a proof can absorb any amount of compute and ingenuity; checking one takes seconds and returns a plain yes or no. Almost no work has that shape, and it is exactly what paying strangers for results requires: the hard part is what earns, and the cheap part is what makes cheating pointless.

It also settles the obvious worry about submitting to an open network: you publish a fingerprint of your file first and the file itself only afterwards, so nobody can watch what you send and claim it as theirs.

How Bittensor subnets work

From the catalog

A few of the open ones.

Each entry states the problem in ordinary mathematical language and gives you the exact Lean statement you would need to prove, along with where it came from. Nothing is paraphrased, so what you read is what gets checked.

Where this is

Proofs and counterexamples both check today. Funding an attempt still needs a human.

  1. Built now

    The pool, the checker, and both directions

    You can browse the published statements, submit a Lean proof, and have it checked against the kernel. Every statement now carries a counterexample task alongside the proof task, and the two are rewarded independently. Each one is pinned to an exact revision of the source repository and to an exact toolchain, so the target cannot move under you.

    See the full process

  2. Coming next

    Funding an attempt without waiting on us

    An attempt is paid for by a transfer to the treasury. Reading that transfer back off the chain is the piece still being built, so a deposit is confirmed by hand rather than the moment it lands. Card and USDC payment is planned after that, so taking part will not require a Bittensor wallet.

    Read the detail in the FAQ

Deliberately limited

One family of problems, and a human before every payout

The pool is drawn entirely from Erdős problems that passed a named audit: still open upstream, no active pull request resolving them, no proof already in the catalogue. Other families are eligible and deliberately unpublished. Rewards are released by hand for the same reason — until there is a real sense of how often solves happen, paying out on a script would be a promise made against a treasury that has not been tested.

See the current numbers