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Math-3029 · mathematics · Topology

Kervaire detection at the prime three

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Which members $b_j$ of the standard prime-three Kervaire-detection family survive to stable sphere homotopy, and which have additive-order-three representatives? The proposed answer is precisely $j=0,2,3$, in stems 10,106,322.

Short English statement

Which members $b_j$ of the standard prime-three Kervaire-detection family survive to stable sphere homotopy, and which have additive-order-three representatives? The proposed answer is precisely $j=0,2,3$, in stems 10,106,322.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3029-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3029

Manuscript 1 (family 309, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Kervaire detection at the prime three

OpenAI's claim

The standard mod-three Adams Kervaire classes are claimed to survive exactly at indices zero, two, and three, each with an order-three representative; the new index-three element lies in stem 322. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
In the mod-three Adams spectral sequence $\operatorname{Ext}_{\mathcal A_3}^{s,t}(\mathbb F_3,\mathbb F_3)\Rightarrow(\pi^S_{t-s})_3^\wedge$, let $b_j$ be the standard Kervaire family in bidegree $(2,4\cdot3^{j+1})$, $j\ge0$. OpenAI claims \[ \{j:b_j\text{ survives nontrivially}\} =\{j:\text{an order-three class is detected by }b_j\} =\{0,2,3\}. \] The surviving stems are $10,106,322$; $b_1$ and all $b_j$, $j\ge4$, fail to survive. Order-three existence is within each detection coset, without asserting that every representative has the same order.

Outline of the argument

Manuscript 1 (family 309, 2026-09-24): A height-six Morava detector with C_9 action imposes finite Euler deadlines and outgoing differentials on all higher candidate classes. Independently, a truncated extended cube of the known stem-106 Moore map carries the required primary operation. Finite obstruction corrections lift it back through the Moore sequence, preserving detection and producing an actual sphere element killed by three.

Formal verification

Family 309 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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