The four-dimensional disc embedding conjecture and whether the free group is good
Freedman's disc embedding theorem says: in an oriented 4-manifold with good fundamental group , immersed discs with embedded boundaries and framed algebraically dual spheres (, in ) can be replaced by disjoint locally flat embedded discs with the same boundary. Groups of subexponential growth are good (Freedman-Teichner, Krushkal-Quinn), and goodness passes to subgroups and quotients, so the decisive open case is the free group , which also governs topological surgery and the 5-dimensional s-cobordism theorem for arbitrary groups. Does the disc embedding conclusion hold without the good-group hypothesis; in particular, is the free group good?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Topological 4-manifolds; surgery and disc embedding
- Posed by
- Michael Freedman and Frank Quinn, Topology of 4-manifolds (1990), theorem for good groups; the free-group case is listed open by Kim-Orson-Park-Ray (2021) and Powell-Ray-Teichner (2025)
- Year posed
- 1990
- Years open
- 36y
- 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
- 60 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a compact oriented smooth 4-manifold with immersed proper discs with embedded boundary circles and framed (even smoothly embedded) spheres satisfying , , , whose boundary circles bound no pairwise disjoint locally flat discs, in any relative homotopy class. Corollary 1.2: , and every group containing it, is not good in the Freedman-Quinn sense. Corollary 1.3: the unrestricted topological surgery and s-cobordism assertions cannot both hold. The marked tensor companion proves the framing-preserving version. Not shown: which of surgery or s-cobordism fails, or that the Borromean rings are not A-B slice.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, 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). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript and the companion 'A marked tensor obstruction' (both September 24, 2026) give two versions of the obstruction; the family's Wall-question manuscript builds on the marked tensor version.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.3 of 'A boundary-only obstruction to four-dimensional disk embedding' were read against the disc embedding theorem as formulated by Powell-Ray-Teichner (2025) and the free-group question. The proof (a capped-grope construction, representation deformation rings with Chern-Simons type wall potentials, reduction to positive characteristic and a finite Frobenius obstruction) was not refereed. No Lean formalization. Scope the paper itself states: it refutes the conjunction of the unrestricted topological surgery and s-cobordism assertions, but does not say which one fails; the counterexample manifold's fundamental group is not identified with F2, and nongoodness of F2 follows via subgroup and quotient closure. A disproof of this standing would be expected to draw expert scrutiny quickly.