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Math-501-A2 · revision r1 · on Math-501

OpenAI claim summary for Math-501

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 310, 2026-09-24): Nonzero augmented reduced rational homology whenever O_p(G)=1 excludes contractibility of the elementary-abelian-subgroup complex, hence proves the conjecture by contraposition for every finite group and prime, with the empty-poset convention included.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Quillen's \$p\$-Subgroup Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Quillen's \$p\$-Subgroup Conjecture

OpenAI's claim

For every finite group and prime with trivial p-core, the elementary-abelian p-subgroup poset is claimed to have nonzero augmented rational homology, proving a stronger homological form of Quillen's conjecture. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a finite group $G$ and prime $p$, let $O_p(G)$ be its largest normal $p$-subgroup and $\mathcal A_p(G)$ the poset of nonidentity elementary abelian $p$-subgroups, ordered by inclusion. OpenAI claims \[ O_p(G)=1\quad\Longrightarrow\quad\widetilde H_*(\mathcal A_p(G);\mathbb Q)\ne0. \] Homology is that of the order complex and uses augmented reduced chains, with $\widetilde H_{-1}(\varnothing;\mathbb Q)=\mathbb Q$. This is rational homological nonvanishing, which implies noncontractibility.

Outline of the argument

Manuscript 1 (family 310, 2026-09-24): Apartment chains from line frames are related by plane changes so that their boundaries cancel. Lie-superbracket encodings and a Poincare--Birkhoff--Witt estimate bound the surviving cycle space while retaining form types and automorphism constraints. These cycles meet the needed component-extension cases, and explicit full-chain propagation passes their nonzero coefficients into the ambient subgroup poset.

Formal verification

Family 310 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