Deformation-stable invariants cannot resolve the Hopf problem: a written record of a folklore obstruction

What this note does. The space of almost complex structures on the six-sphere inducing a fixed orientation is connected. This note draws the consequence: every invariant of almost complex structures on S that is constant on homotopy classes and natural under diffeomorphism is constant outright. Hence no such invariant can distinguish a hypothetical integrable structure from the octonionic J₀, and no obstruction scheme built from characteristic classes, symbol classes in K-, KO-, KSp- or KR-theory, indices extracted from them, or bordism can resolve the Hopf problem negatively. The note states this as a theorem, applies it as an audit criterion, and delimits the argument types that escape it. The sharpest consequence, for Atiyah's 2016 note (arXiv:1610.09366). That argument rests on three assertions: (A) even/odd type is a topological property shared by all almost complex structures; (B) the octonionic J₀ is odd; (C) an integrable structure would be even. Granting (A) and (B), the no-go theorem forces the type to be constantly odd, and assertion (C) then holds if and only if S admits no complex structure. So (C) is not a lemma from which non-existence can be deduced — it is logically equivalent to the conclusion, and the deduction is circular rather than merely incomplete. This survives even if every withheld definition in that paper were supplied and every computation verified. It is a different and sharper criticism than the published one, which is that the definitions are missing. Attribution, stated plainly. The topological input is classical and is not this note's. Bryant states it with proof in arXiv:1405.3405 §3.3 (2014), two years before the paper audited here, via the same polar-decomposition retraction; Tang and Yan restate it obstruction-theoretically in arXiv:2601.05568, Remark 4.1 (2026); Ferlengez, Granja and Milivojevic take it as known. The objection to Atiyah's argument is likewise not new as an observation: it exists as public expert folklore, in particular Bryant's remark that the argument read literally would prove S admits no almost complex structure, which is false, and Wieland's isolation of the reason, that the operator's symbol depends only on the almost complex structure. Both are cited to the MathOverflow threads where they appear. What appears to be unrecorded, in print or in public discussion, is the closure — that this is a theorem about the entire class of such schemes rather than a defect of one paper — together with the circularity diagnosis above. Status and scope. This is a preprint, not a peer-reviewed article, and it has not been read by a human mathematician other than the author. It should be treated accordingly. The mathematics is elementary: the main theorem follows from the classical connectedness statement in three lines. The note does not claim the Hopf problem is hard, and it does not bear on whether S admits a complex structure; it says only where the difficulty is not. Two claims are deliberately hedged rather than asserted: whether the theorem applies to Adler's 1969 paper is left open, since that paper has not been read here and specialists report its argument is metric-dependent, in which case the theorem does not apply. Verification. Two independent AI verification passes were run, and both found genuine errors, which are recorded in the note itself rather than silently fixed. The first found that an earlier draft omitted the naturality condition and was wrong by omission, since a hypothetical integrable structure could inhabit the opposite orientation component. The second found that an earlier form of the main corollary asserted that no invariant satisfies (A), (B) and (C) simultaneously — which is false, and was inadvertently equivalent to an affirmative solution of the Hopf problem. The algebraic identity underlying the polar-decomposition retraction was checked in exact rational arithmetic and the retraction verified numerically on random almost complex structures. Use of AI assistance. This note was drafted and literature-checked with substantial assistance from AI language models (Anthropic Claude), including the formulation of the statements and the proofs as written. Neither verification pass is a substitute for a specialist read, which has not happened. The author is responsible for the content. This disclosure is made unconditionally.