The Target-Free Clique Conjecture for Threshold-Linear Networks
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge from every member of. An explicit six-neuron counterexample refutes it.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Mathematical neuroscience
- Posed by
- Carina Curto, Jesse Geneson, Katherine Morrison
- Year posed
- 2019
- Years open
- 7y
- Solved
- 2026-07-23
- Model
- Codex (GPT-5.6), Claude Code (Opus 4.8, Fable 5)
- Vendor
- OpenAI / Anthropic
- Collaborators
- Jesse Geneson
- Verification
- Site-confirmed
- Publication
- Preprint
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The acknowledgement says the systems were used for proof exploration, proof criticism, exposition and revision, with no specific step attributed, so the lowest tier applies.
Verification
The counterexample is a single fixed six-vertex graph with an explicit parameter regime, so the refutation reduces to a finite determinant and clique computation set out in the paper. Single-author arXiv preprint, not yet peer-reviewed.
Source
arXiv:2607.21396 - A six-neuron counterexample to the target-free clique conjecture