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Math-2382-A1 · revision r1 · on Math-2382

OpenAI claim summary for Math-2382

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: open. 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 299, 2026-09-25): The character criterion answers the central-sequence branch for nonzero unital separable A, and the infinite-power corollary handles separable D without characters; the extracted bundle omits these restrictions, so unrestricted nonseparable central-sequence absorption is not inferred.

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

Manuscript

Contents
  1. Jiang–Su Embeddings in Character-Free Central Sequence Algebras
  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

Jiang–Su Embeddings in Character-Free Central Sequence Algebras

OpenAI's claim

For nonzero unital separable C*-algebras, absence of characters in the norm central sequences is claimed to characterize Jiang--Su absorption; characterless algebras also have Jiang--Su-stable infinite minimal tensor powers. 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 nonzero unital separable complex $C^*$-algebra $A$ and free ultrafilter $\omega$, define its norm ultrapower $A_\omega$ and central-sequence algebra $F_\omega(A)=A_\omega\cap A'$, using the diagonal copy of $A$. OpenAI claims \[ F_\omega(A)\text{ has no nonzero }*\text{-homomorphism to }\mathbb C \quad\Longleftrightarrow\quad A\simeq A\otimes_{\min}\mathcal Z, \] where $\mathcal Z$ is the Jiang--Su algebra. It additionally claims that for every nonzero unital separable characterless $D$, its infinite minimal tensor power $D^{\otimes_{\min}\infty}=\varinjlim(D^{\otimes n},x\mapsto x\otimes1)$ absorbs $\mathcal Z$ tensorially and admits a unit-preserving inclusion of $\mathcal Z$.

Outline of the argument

Manuscript 1 (family 299, 2026-09-25): Finite tensor labels detect noncommutativity in every representation and build a full square-zero element. Its two orthogonal corners are exactly equivalent, and a finite comparison absorbs their unit defect into a positive cutoff. This supplies dimension-drop maps; reindexed commuting copies then pass from the finite tensor construction to central-sequence absorption and infinite tensor powers.

Formal verification

For family 299, the scope page documents the following selections. The selection equates Jiang--Su absorption with absence of characters in the norm central-sequence algebra for nonzero unital separable algebras, without nuclearity or trace assumptions. The tensor-power consequence is separate. The documented statement covers the main claim of manuscript 1. It does not establish formal coverage of the complete problem clause. Comparator files: CharacterCriterion.lean. Declarations: OAI.KirchbergRordam.character_criterion.

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