Bellman's Lost-in-a-Forest Problem for the Golden Gnomon
What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides and apex angle - from an unknown starting position and heading? The optimum is a symmetric seven-piece path of segments, circular shoulders and tangents, of exactly determined transcendental length - the first proved exact optimum for an isosceles triangle with base angle below .
- Result
- Proved (Bellman's problem for general regions remains open)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Convex geometry
- Posed by
- Richard E. Bellman
- Year posed
- 1956
- Years open
- 70y
- Solved
- 2026-07-27
- Model
- Claude Fable 5, GPT-5.6 Sol, Claude Opus 5
- Vendor
- Anthropic / OpenAI
- Collaborators
- Alexander Temerev, Alessio Doria
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- 1 language
What the AI did
The models were used throughout: to search out the extremal curve, to draft the arguments, and to write the accompanying Lean 4 development. The paper states precisely which steps are machine-checked, and notes those checks hold regardless of how the statements were found.
Verification
Lean 4 verifies the two finite algebraic certificate families and the discrete ledger identities, but not the full argument end to end; the paper's appendix states exactly which steps are machine-checked. arXiv preprint, not yet peer-reviewed.
Sources
arXiv:2607.24483 - The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon