Levent Alpöge (Harvard), working with Claude, circulated a ~108-page PDF claiming a construction of an integrable complex structure on S⁶ — the Hopf problem, open since 1947. As of market creation the paper is circulating as a PDF outside arXiv; his most recent arXiv submission is "More than two thirds of the zeta zeros are simple and on the critical line" (2026-08-13), and his July 2026 Jacobian-conjecture counterexample is not on arXiv either.
This market is about the posting, not the correctness.
Resolves YES if, at any time from market creation through 2026-10-15, 23:59 ET, arxiv.org lists a preprint (any version, any primary category) with Levent Alpöge as an author whose subject is the existence of a complex / integrable complex structure on the 6-sphere.
Resolves NO otherwise — including if the paper only ever circulates as a personal-website or social-media PDF, is submitted to a journal without an arXiv posting, is withdrawn, or appears on arXiv without Alpöge as an author.
Oracle: the arXiv listing itself — https://arxiv.org/a/alpoge_l_1 and arXiv full-text search. A cross-listing or replacement counts; an announcement of intent to post does not. Timestamp used is arXiv's own submission date.
Correctness of the proof is explicitly out of scope. A paper that is posted and later refuted still resolves this YES.
1,000Ho ho ho — opened it at 55% because that's honestly where I have it.
Checked his list before making this one: he arXivs constantly (latest was the zeta-zeros paper on Aug 13, arxiv.org/a/alpoge_l_1) — but the July Jacobian counterexample, his most famous result of the year, still isn't on there. Active poster, and yet the big splashy thing stays a PDF. That's the whole coin flip.
Correctness is not what's being asked here. Posted and later refuted still resolves YES.
The cycle continues.