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Math-2716-A1 · revision r1 · on Math-2716

OpenAI claim summary for Math-2716

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: 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.

The Kervaire corollary states precisely nontriviality for every w when A is nontrivial. The main unimodular injectivity theorem handles degree plus/minus one; the other degrees retain a nontrivial cyclic quotient. lean/docs/256.md documents the selected unimodular injectivity theorem, not this all-word corollary as an exact formal statement. Within these assumptions, this is a claimed affirmative answer to the stated question.

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

Manuscript

Contents
  1. A nontrivial group free-product Z cannot be normally generated by one element
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

A nontrivial group free-product Z cannot be normally generated by one element

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For every nontrivial group $G$, the free product $G*\mathbb Z$ is claimed not normally generated by a single element.

Mathematical statement of the claimed result

Result 1.
In the free product $H=A*\langle t\rangle$, a word $w$ is unimodular when its total exponent of $t$ equals $1$ or $-1$. For any coefficient group $A$ and such $w$, the source claims \[ A\longrightarrow H/\langle\!\langle w\rangle\!\rangle_H\text{ injective}. \] The Kervaire corollary drops the exponent assumption for a different conclusion: \[ A\ne1\quad\Longrightarrow\quad H/\langle\!\langle w\rangle\!\rangle_H\ne1\qquad(w\in H). \]

Outline of the argument

A traced spectral phase provides a unitary with a controlled large fixed space. Translating a putative coefficient relation into a planar surface yields one nontrivial boundary, which the phase calculation excludes; this prevents a normal generator of the required form.

Formal verification

For family 256, the selected statement is coefficient injectivity for a single relator of exponent sum $1$ or $-1$ in $A*\mathbb Z$. It underlies the Kervaire consequence but does not include the newer general nonsingular-system theorem. The listed Comparator files are Kervaire.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources