Resolution criteria

This market will resolve to YES if the purported construction of a complex structure on $S^6$ by Levent Alpöge is verified and widely accepted by the mathematics community.

"Verification and wide acceptance" is defined as meeting at least one of the following conditions:

1. Journal Publication: The paper (or a revised version of it presenting the same fundamental construction) is published in a peer-reviewed mathematics journal of high standing.

2. Community Consensus: Leading experts reach a clear consensus that the construction is correct and free of fatal gaps.

This market will resolve to NO if:

1. Fatal Error Found: A fatal, unfixable error or gap in the proof is identified and widely accepted by the mathematics community.

2. Withdrawal: Levent Alpöge publicly retracts the preprint or acknowledges a fatal, uncorrectable error in the construction.

3. Cutoff Reached: The cutoff date of December 31, 2028 (or another date edited by the creator) is reached without the proof meeting the criteria for YES.

Minor, fixable errors or typos that are successfully resolved by the author will not trigger a NO resolution. In the event of ambiguity, the market creator will use their best judgment based on the consensus of the professional mathematics community.

Background

The question of whether the 6-dimensional sphere ($S^6$) admits a complex structure is one of the most famous open problems in differential geometry, dating back to the mid-20th century.

By a classic result of Armand Borel and Jean-Pierre Serre, the only spheres that admit an almost complex structure are $S^2$ and $S^6$. While the almost complex structure on $S^2$ is integrable (making it the Riemann sphere), whether the almost complex structure on $S^6$ can be integrated into a true complex structure has remained open. Multiple high-profile mathematicians (including Michael Atiyah in 2016) have announced proofs or constructions over the years, all of which were subsequently found to have fatal gaps.

On August 23, 2026, mathematician and Anthropic researcher Levent Alpöge posted a preprint titled "A complex structure on $S^6$" on his personal website (alpo.ge). Alpöge has recently gained significant attention for collaborating with large language models (specifically Anthropic's Claude) to resolve major open mathematical questions, such as disproving the Jacobian Conjecture in July 2026. This market tracks whether his newly proposed construction of a complex structure on $S^6$ will withstand rigorous peer review.

To my knowledge, Alpöge has not directly claimed any AI involvement in the S^6 construction, but I and probably most others expect the proof is at least in part and possibly entirely by Claude. AI authorship or assistance does not affect the outcome of this market.