No bigeodesics in planar first-passage percolation under a minimum-of-four second-moment condition
Give the nearest-neighbor edges of i.i.d. nonnegative weights with a continuous law and let be the passage-time metric. A bigeodesic is a doubly infinite path every finite segment of which is a geodesic. Furstenberg asked, as recorded by Kesten (1986, Remark 9.22), whether such paths can exist; the planar no-bigeodesics conjecture (Damron-Hanson, Conjecture 1) asserts that almost surely there are none for i.i.d. continuous weights. Earlier results excluded bigeodesics only in fixed directions, under curvature or differentiability assumptions on the limit shape, or in restricted geodesic graphs. Is it true that almost surely planar first-passage percolation with i.i.d. continuous weights has no bigeodesic?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- First-passage percolation; infinite geodesics
- Posed by
- Hillel Furstenberg, as recorded by Harry Kesten; stated as a conjecture by Damron and Hanson (2017)
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if the edge-weight law on has no atoms and the minimum of four independent copies has finite second moment, then almost surely there is no bigeodesic, all paths excluded simultaneously on one probability-one event, with no assumption of differentiability, strict convexity or curvature of the limit shape. The law may be singular continuous, have bounded support or infinite mean. It does not cover laws failing the moment condition, higher dimensions, or directed models.
What the AI did
The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the manuscript states it. It assumes nonatomic weights with and the paper says explicitly that it does not assert the moment-free formulation; hence Partial. The structural inputs (Ahlberg-Hoffman's Busemann theory) are cited theorems from the literature, replaced at one step by an appendix argument. The proof was not refereed. This manuscript has no Lean formalization; the family's Lean work (lean/docs/212.md) covers only the companion limit-shape paper.