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Math-2204-A1 · revision r1 · on Math-2204

OpenAI claim summary for Math-2204

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 306, 2026-09-23): Theorem 1.1 proves clause (c) for every nontrivial S3 knot, but the bundle also asks cosmetic/complement questions in arbitrary closed three-manifolds and a chirally cosmetic classification, so the bundle is not wholly resolved.

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

Manuscript

Contents
  1. Kirby Problem 1.12
  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

Kirby Problem 1.12

OpenAI's claim

For any nontrivial smooth knot in $S^3$, the paper claims that surgeries at different slopes cannot be orientation-preservingly homeomorphic, resolving purely cosmetic surgery. 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 any smooth nontrivial knot $K\subset S^3$ and slopes $r,s\in\mathbb Q\cup\{\infty\}$, OpenAI claims \[ S^3_r(K)\cong S^3_s(K)\text{ by an orientation-preserving homeomorphism}\ \Longrightarrow\ r=s. \] Slopes use the meridian and preferred longitude; $p/q$ represents the primitive class $p\mu+q\lambda$ up to simultaneous sign. The orientation of the Dehn filling extends the knot exterior's orientation. This assertion concerns purely cosmetic surgery on a single knot in $S^3$, excluding the unknot and orientation-reversing equivalences.

Outline of the argument

Manuscript 1 (family 306, 2026-09-23): Existing Floer results confine a possible cosmetic pair to slopes +2 and -2 with knot genus two. Integral determinant lifts produce a nonzero instanton count, while toric period and tube estimates exclude unwanted reducible and broken strata. Compatible spinor regularization then builds an oriented one-dimensional count whose allowed boundary would force a nonzero multiple of that instanton count to vanish.

Formal verification

Family 306 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