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Math-2566-A1 · revision r1 · on Math-2566

OpenAI claim summary for Math-2566

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.

Manuscript 1 (family 064, 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.

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

Manuscript

Contents
  1. Topological right-triviality of constant-Milnor surface families
  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

Topological right-triviality of constant-Milnor surface families

OpenAI's claim

A one-parameter isolated surface-hypersurface family with constant Milnor number is claimed to admit a jointly continuous ambient trivialization preserving both parameter and function value. 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 holomorphic family $F:(\mathbb C^3\times\Delta,\{0\}\times\Delta)\to(\mathbb C,0)$, put $f_t(x)=F(x,t)$. Assume each $f_t(0)=0$ has an isolated critical point and constant Milnor number \[ \mu(f_t,0)=\dim_{\mathbb C}\mathcal O_{\mathbb C^3,0}/(\partial_1f_t,\partial_2f_t,\partial_3f_t)=\mu(f_0,0). \] OpenAI claims, after shrinking the disk and representatives, a parameter-preserving homeomorphism germ $\Phi(x,t)=(\phi_t(x),t)$ with $\phi_t(0)=0$, $\phi_0=\mathrm{id}$ and $f_t(\phi_t(x))=f_0(x)$. Both $\Phi$ and its inverse are jointly continuous along the origin section. This is topological right-triviality, including defining functions rather than only zero sets.

Outline of the argument

Manuscript 1 (family 064, 2026-09-24): Semistable reduction and a log program compare extra boundary surfaces by compact-support Euler characteristics. Constant Milnor number cancels the main contribution, and nonnegative smoothing excesses plus the orbifold surface inequality force their log squares to vanish; resolution transport then extends to the ambient function levels.

Formal verification

Family 064 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