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Math-211-A3 · revision r1 · on Math-211

OpenAI claim summary for Math-211

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 304, 2026-09-23): Theorems 1.1–1.2 settle the stated unrestricted homeomorphism action problem; for an arbitrary locally compact acting group, faithfulness embeds each compact neighborhood into the second-countable homeomorphism space, so a generated open subgroup is second countable and Lie, and its translates give the compatible Lie structure with arbitrary component count.

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

Manuscript

Contents
  1. Hilbert-Smith 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

Hilbert-Smith conjecture

OpenAI's claim

A faithful continuous action of a second-countable locally compact group on any connected finite-dimensional topological manifold is claimed to force the group to be Lie, resolving Hilbert--Smith in every finite dimension. 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 every finite integer $n\ge1$, OpenAI asserts that a Hausdorff second-countable group $G$, locally compact, with a faithful action on a connected topological $n$-manifold $M$. Require $M$ to be Hausdorff, second-countable and without boundary, and the action map $G\times M\to M$ to be continuous. Under these conditions $G$ is a Lie group. Its $p$-adic exclusion theorem says \[ \ker(\mathbb Z_p\curvearrowright M)\ne\{0\},\qquad \mathbb Z_p=\varprojlim_k\mathbb Z/p^k\mathbb Z, \] for every prime $p$ and every such jointly continuous action; therefore some finite quotient already determines the action. Faithfulness means trivial global kernel, and does not mean free action.

Outline of the argument

Manuscript 1 (family 304, 2026-09-23): Witt groups of sheaves generated by continuous polyhedral maps acquire a fixed integral signature lattice through cutoff localization and doubling comparisons with semialgebraic objects. A hypothetical p-adic action produces equally valued classes in arbitrarily large orbit batches. Their rational values would have unbounded p-power denominators, contradicting the lattice fixed before the batch size is chosen.

Formal verification

Family 304 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