Exactly one half?
Two AI preprints claim the radius of comparison is half the mean dimension, the Phillips–Toms conjecture
Covers result 302 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
3:57. English captions available. The video loads from YouTube when you press play.
Chapters
The claims, in plain words
Mean dimension, introduced by Gromov and developed by Lindenstrauss and Weiss, measures how many real dimensions a dynamical system gains per time step. For the shift on sequences of points in a d-dimensional cube it is d. It is not entropy: the coin-flip shift has entropy log 2 but mean dimension 0. The radius of comparison, introduced by Toms in 2006, measures how far a C*-algebra is from strict comparison. For a minimal homeomorphism, the crossed product links the two, and Phillips and Toms conjectured that its radius of comparison is half the mean dimension, a factor suggested by vector bundles: a complex bundle of rank r has real dimension 2r. Niu proved the upper bound, and Hirshberg and Phillips had lower bounds that, for certain special subshifts, come within one of the conjectured value.
Result 302 is two OpenAI papers, 49 pages, dated September 25, 2026, and published October 6. The main paper claims that for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of the crossed product equals half the mean dimension, including the case where both are infinite; the new part is the lower bound. Its corollary: zero mean dimension, the small boundary property, 𝒵-stability, finite nuclear dimension and nuclear dimension at most one are all equivalent. Most links were known; the new one is that 𝒵-stability, or finite nuclear dimension, forces mean dimension zero, which Elliott and Niu had claimed in 2025 before a gap was found.
What they don’t claim
Anything about actions of groups other than the integers, or about non-minimal systems.
Where it stands
- Posted as preprints
- OpenAI, two papers, 49 pages, dated September 25, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No formalization for this item
Last checked October 19, 2026. Neither paper has been revised, and there is no Lean formalization for this item. OSVerify has not reviewed it. On October 8, Elliott and Niu posted a follow-up, carried out using ChatGPT, that uses the cobordism theorem as its main technical ingredient and extends the equality to some amenable group actions. It raises no doubts, but it is not a check, and we found no other expert comment.
How the proof goes
- An engine from algebraic topology. The main paper turns a comparison bound into a bundle embedding of less than twice the rank. The companion paper’s compression theorem in complex cobordism, built on quantitative forms of the Devinatz–Hopkins–Smith nilpotence theorem due to Mathew and Burklund, turns that embedding into boundary slots in a coloured cover of orbit blocks. Each boundary slot deletes a colour, so the cover becomes thinner than its growth rate allows: a contradiction.
- The 1/2, and the budget. The 1/2 appears as the limit of p/(2p + 2). With a comparison bound of 0.4 against a cover rate of 1, the smallest packets have 12 colours, and each window must hold over 100,000 orbit blocks (our computation of the paper’s inequalities).
The numbers in the video
- Ours: for the cube shift, the best refinement of a natural cover of an N-step window has order exactly dN, matching the known mean dimension d.
- Ours: a classic Chern-class obstruction, going back to Villadsen: a suitable line bundle on a product of k two-spheres needs a complement of rank at least k, which forces a radius of comparison of at least k − 1.
- Ours: irrational rotations and odometers have entropy 0 and mean dimension 0; the coin-flip shift has entropy log 2 and mean dimension 0.
Read the source
- Radius of Comparison Equals Half the Mean Dimension (PDF), 24 pages.
- Filtered Products and Boundary-Preserving Compression in Complex Cobordism (PDF), 25 pages.
- Elliott and Niu’s follow-up (arXiv, October 8, 2026).
- The collection’s catalog, where this is result 302.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Mean dimension, introduced by Gromov and developed by Lindenstrauss and Weiss, measures how many real dimensions a dynamical system gains per time step. Our computer checks the model case: for the shift on sequences of points in a d-dimensional cube, the best refinement of a natural cover of an N-step window has order exactly dN, matching the known mean dimension, d. It is not entropy: the coin-flip shift has entropy log 2 but mean dimension 0, and irrational rotations and odometers have both equal to 0. The radius of comparison, introduced by Toms in 2006, measures how far a C*-algebra is from strict comparison. For a minimal homeomorphism, the crossed product links the two worlds, and Phillips and Toms conjectured that its radius of comparison is half the mean dimension. Niu proved the upper bound: the radius is at most half the mean dimension. The factor 1/2 is suggested by vector bundles: a complex bundle of rank r has real dimension 2r. Our computer checks a classic Chern class obstruction, going back to Villadsen: a suitable line bundle on a product of k two-spheres needs a complement of rank at least k, which forces a radius of comparison of at least k − 1, half the dimension minus one. Hirshberg and Phillips had lower bounds, and for certain special subshifts these come within one of half the mean dimension.
1:29The claims
Result 302 is two OpenAI papers, 49 pages, dated September 25 and published October 6, 2026. The main paper claims that for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of the crossed product equals half the mean dimension, including the case where both are infinite. The new part is the lower bound. Its corollary: zero mean dimension is equivalent to the small boundary property, 𝒵-stability, finite nuclear dimension, and nuclear dimension at most one. Most links were known; the new one is that 𝒵-stability, or finite nuclear dimension, forces mean dimension 0, which Elliott and Niu had claimed in 2025, before a gap was found; by August 2026 they called it open. Not claimed: actions of groups other than the integers, or non-minimal systems.
2:26How the proof goes
The companion paper supplies the engine: a compression theorem in complex cobordism, built on quantitative forms of the Devinatz–Hopkins–Smith nilpotence theorem due to Mathew and Burklund. The main paper turns a comparison bound into a bundle embedding of less than twice the rank. The compression theorem turns that embedding into boundary slots in a coloured cover of orbit blocks. Each boundary slot deletes a colour, so the cover becomes thinner than its growth rate allows: a contradiction. The 1/2 appears as the limit of p/(2p + 2). Our computer works out the paper’s parameter budget: with a comparison bound of 0.4 against a cover rate of 1, the smallest packets have 12 colours, and then each window must hold over 100,000 orbit blocks.
3:17Where it stands
Both papers are unrefereed and unchanged since release, and there is no Lean formalization for this item. On October 8 Elliott and Niu posted a follow-up, carried out using ChatGPT, which uses the cobordism theorem as its main technical ingredient and extends the equality to some amenable group actions. It raises no doubts, but it is not a check, and we found no other expert comment. If it holds, for minimal homeomorphisms the radius of comparison is exactly half the mean dimension, and regularity of the crossed product is the same as mean dimension 0.
Spot a mistake?
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