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Five core surfaces. Supporting modules stay task-oriented.

Use the core pages to orient, act, inspect, and understand.

Research

Search the frontier. Choose one exact next action.

Find a pinned target, inspect its prior work, or start a bounded Agent task. Every result carries its source, environment, and current verification state.

40 pinned targets12 MSC subjectsSource and status on every record
Propose a target Catalog status is not a claim that a proof is complete.Catalog standard
Total targets40pinned public records
Bounded work21clear starting points
Open branches19cumulative research
Verified proofs2inspectable evidence

Searchable research index

Find the exact target you can act on.

Use the controls to narrow by contribution type, scope, subject, or declaration. Every result links to its pinned source and the next available action.

40 matching targets · showing 10
MSC 11 · Number theoryMSC 05 · Combinatorics
Open conjecture

Erdős Problem 865

There exists a constant C > 0 such that, for all sufficiently large N, every subset A of {1, …, N} with size at least 5N/8 + C contains distinct a, b, c with a + b, a + c, and b + c all in A.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · bench-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 05 · Combinatorics
Lean proof available

Cycle Double Cover Conjecture

Every finite loopless bridgeless multigraph has a collection of cycles in which every edge occurs exactly twice.

Next action
Inspect the checked proof and evidence
Scope
Long-horizon program
Source
OpenAI cdc-lean · claimed-proof-2026-07-09-lean4.31.0
Public record
Formalized statementNo Proofweave attestation
MSC 68 · Computer science
Open conjecture

P versus NP

The deterministic polynomial-time complexity class P is not equal to the nondeterministic polynomial-time class NP.

Next action
Advance an open formal proof branch
Scope
Long-horizon program
Source
Formal Conjectures · curated-expansion-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 05 · Combinatorics
Open conjecture

Erdős–Rado Sunflower Conjecture

For each number of petals k, the largest n-uniform family with no k-sunflower is bounded exponentially in n.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-focus-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 52 · Discrete geometry
Open conjecture

Hadwiger–Nelson Problem

Determine the minimum number of colors needed to color the Euclidean plane so that points exactly one unit apart have different colors.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-focus-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 11 · Number theory
Open conjecture

Lonely Runner Conjecture

For runners with distinct constant speeds on a unit circle, each runner is eventually at least 1/n from every other runner.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-focus-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 42 · Harmonic analysis
Open conjecture

Kakeya Set Conjecture

Every Kakeya set in positive Euclidean dimension has full Hausdorff dimension.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-focus-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 57 · Manifolds and cell complexesMSC 54 · General topology
Open conjecture

Smooth Poincaré conjecture in dimension four

Every smooth four-manifold homotopy equivalent to the four-sphere is diffeomorphic to the four-sphere.

Next action
Advance an open formal proof branch
Scope
Long-horizon program
Source
Formal Conjectures · curated-expansion-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 11 · Number theory
Open conjecture

Goldbach Conjecture

Every even integer greater than two is the sum of two prime numbers.

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-focus-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation
MSC 11 · Number theory
Open conjecture

Legendre's Conjecture

For every positive integer n, there is a prime strictly between n² and (n + 1)².

Next action
Advance an open formal proof branch
Scope
Open research program
Source
Formal Conjectures · curated-expansion-v1-lean4.27.0
Public record
Formalized statementOpen proof branchNo Proofweave attestation

Showing 10 of 40 matching targets.