Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.
The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.
In The Kervaire invariant problem at the prime three, 24 September 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
At the prime three,
Thus the surviving classes occur in stems 10, 106, and 322, and each admits an additive-order-three representative. The class b_1 and every b_j with j\geq4 fail to survive.
In The Kervaire invariant problem at the prime three, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There are homotopy equivalences of H-spaces
Consequently, W_{81}\simeq\Omega^2T^{487}(3). • For every integer n>1, the underlying space \Omega S^{2n+1}\{3\} has a nontrivial product decomposition if and only if n\in\{3,27,81\}. For every other n>1 it is mod-three atomic, hence indecomposable. Here a product is nontrivial when both factors are noncontractible. • For every integer n>1, the homotopy type T^{2n+1}(3) admits a homotopy-associative H-space multiplication if and only if n\in\{3,27,81\}. The multiplication on T^{163}(3) supplied by the displayed product is homotopy associative and homotopy commutative. For this multiplication, the third-power map on T^{163}(3) is null-homotopic, as is the third-power map on the loop H-space \Omega T^{487}(3).
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change
The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
The Kervaire invariant problem at the prime three, 24 September 2026
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Offer to write an explanationSuggest a source attributionSuggest a change
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In The Kervaire invariant problem at the prime three, 24 September 2026
Why this passage is shown
Corollary 1.2
In The Kervaire invariant problem at the prime three, 24 September 2026
Report using another proof or calculation method
The Kervaire invariant problem at the prime three, 24 September 2026
Exact calculations for finite claims, cert-machine
9814e4c