The Hilbert–Smith conjecture in every finite dimension

Follow-up works

• A. Dabrowski's AI agents, Integral tail signatures and the Hilbert–Smith conjecture (GitHub repository adbrw/HS_proof, paper dated 2026-10-08; PDF). A separate claimed proof, presented by its author as independently developed, that every continuous $\mathbb{Z}_p$-action on a connected finite-dimensional manifold has nontrivial kernel, using integral signature tails and controlled L-theory; it comes with a Lean 4/Mathlib formalization with Comparator and SafeVerify scripts, and the repository page does not mention the OpenAI paper. [AI-generated per source]
• Alan J. Coppola, p-Adic Transfer, Signature Tails, and Homology Manifolds: From a 1980 Binghamton Thesis to the 2026 Hilbert–Smith Arguments (public GitHub draft, 2026-10-18). Compares the OpenAI and Dabrowski arguments and proposes extensions to homology manifolds, marking results that depend on acceptance of the 2026 manifold theorems as relative. [AI-generated per source]

Comments and discussion

• Alan J. Coppola, hilberts-5th-oai: Sherlock reverse-verification of OpenAI Hilbert-Smith preprint (Sept 2026) (GitHub repository, checkpoints 2026-10-18/19; audit ledger). A lemma-by-lemma audit of this paper and the Dabrowski preprint; it reports no FAIL grades for this paper but WEAK grades for the dense doubling step, the Balmer–Walter bridge and the equal-parts isometry, with character coproducts also flagged in the ledger. [AI-generated per source]
• Alex Elzenaar, Answer to “How to complete an AI generated verification of an OpenAI proof for the Hilbert-Smith Conjecture” (MathOverflow, 2026-10-07). Responding to an AI-assisted audit of this paper, Elzenaar calls for expert understanding and sustained mathematical review rather than treating an AI-generated verification as sufficient.

Formalization

• Deicyde, hilbert-smith: Lean 4 formalization of the OpenAI proof of the Hilbert-Smith conjecture in every finite dimension (GitHub, 2026; blueprint). An in-progress Lean 4 formalization that follows this preprint ("oai-math #304"); at the time of collection the blueprint reported "5 of 57 targets complete". It was developed with AutoformBot.
• mo271, HilbertProblems/5: the Hilbert–Smith conjecture is true (google-deepmind/formal-conjectures issue #6914, 2026-10-07). Proposes marking hilbert_smith_conjecture and hilbert_smith_padic_formulation as solved on the basis of this paper, noting that the release contains no Lean proof and that the formal statements do not assume second countability, so this mismatch must be resolved first. The issue says it was prepared with AI assistance.
• adbrw, Hilbert–Smith: add second-countable variants with a formal proof (google-deepmind/formal-conjectures PR #7043, opened 2026-10-09, open). Adds second-countable variants of the formal-conjectures statements, with formal_proof links to the independent adbrw/HS_proof formalization rather than to this paper; a reviewer reports reproducing the Comparator check, and the main statements stay research open.

Related prior literature mentioned by others

• Jon Woolf, Witt groups of sheaves on topological spaces (Commentarii Mathematici Helvetici 83, 2008). Coppola's AI-assisted audit identifies Woolf's interval argument as the antecedent of the paper's homotopy-invariance construction and highlights its passage to the category generated by arbitrary continuous polyhedral images.

Possibly relevant literature (found by the agent, not checked)

• Alexander Dranishnikov, On some problems related to the Hilbert-Smith conjecture (arXiv:1504.07277, v3 2016). Reduces the conjecture to problems in algebraic topology when the orbit space is finite-dimensional, a hypothesis the paper explicitly does not assume ("No dimension bound on the orbit space is assumed").
• Michael Levin, On p-adic actions raising dimension by 2 (arXiv:1910.01476, 2019). Characterizes which $(n+2)$-dimensional compacta arise as orbit spaces of $p$-adic actions on $n$-dimensional compacta, which bears on the Yang / Bredon–Raymond–Williams dimension-raising theory discussed in the paper's introduction.
• Terence Tao, The Hilbert-Smith conjecture (blog post, 2011-08-13). An expository account of the reduction to $\mathbb{Z}_p$ via Gleason–Yamabe theory and Newman's theorems, which the paper uses, citing Lee 1997 and Pardon 2013 for it.

Digests and expositions

• Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, submitted 2026-10-12). Section 8.2.4 is a card on this paper listing its new instruments (compact-source test categories, a fixed sphere lattice, character classes) and reused ingredients, and flagging steps that "require closer examination", such as whether the Balmer–Walter hypotheses pass to localizations of the generated categories. [AI-generated per source]
• Gradient Damage, Hilbert-Smith Conjecture: OpenAI Claims a Proof in Every Dimension (video and accompanying exposition, 2026-10-12). A short presentation that links the paper and illustrates the $p$-adic setting with a 3-adic solenoid animation, described as an illustration rather than a proof.