Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six
Let be the closed ball of capacity (capacity times squared radius) in . Two obstructions to embedding symplectically into the interior of are known: volume, , and Gromov's two-ball obstruction, . In dimension four further obstructions exist. Siegel and Yao conjectured (Conjecture A) that in dimension these two are the only ones. For , is every finite family of closed balls satisfying both inequalities symplectically embeddable into the open ball?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Symplectic geometry, embedding problems
- Posed by
- Kyler Siegel and Yuan Yao
- Year posed
- 2025
- Years open
- 1y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for , and positive , the closed balls embed disjointly and symplectically into if and only if and for . The construction uses a positive-genus surface base with toric fibres and a Hamiltonian comparison lemma. Not shown: anything in dimension four, where more obstructions exist, or the stabilized packing problems Siegel and Yao also study.
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 23, 2026).
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Conjecture A as the manuscript cites it; the theorem states the conjectured criterion as an equivalence for all , and positive capacities. Lean: the challenge BallPacking (theorem OAI.HigherDimensionalBallPacking.main_theorem, solution module OAI.Geometry.BallPacking.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. It defines smooth symplectic embeddings on open neighbourhoods of the closed balls with disjoint images in the open target ball and states the iff with the two inequalities, which is the headline. Not rebuilt here.