What happened
Whether S^6 admits an integrable complex structure is Hopf’s 1948 question. Alpöge’s manuscript constructs a 1-parameter family of compact complex threefolds from a (3,4,∞) torus family and claims they are diffeomorphic to S^6. Philip Engel later wrote a human exposition of the geometric ideas. The construction appears to contradict a 1998 algebraic-dimension obstruction, which is one reason the community has not accepted it. Formal Conjectures still lists the Hopf problem as open. Credit is to Claude under Alpöge’s direction; the public remarks do not pin a Fable/Opus SKU, so identity stays in the Claude family as a named Fable-era collaboration only if later sources specify it. Until then this record uses version-unspecified Claude, not Fable 5 by default.