V(4), at last?
Two AI preprints claim finite Smith–Toda complexes at every height, and V(4) at the prime 1009
Covers result 308 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
4:26. English captions available. The video loads from YouTube when you press play.
Chapters
The claims, in plain words
In stable homotopy theory, Brown–Peterson homology at a prime p has coefficients a polynomial ring in classes v₁, v₂, …, with vₙ in degree 2(pⁿ − 1), and with v₀ standing for p itself. A Smith–Toda complex V(n) is a finite spectrum whose Brown–Peterson homology is that ring with p and v₁ through vₙ killed, built from 2ⁿ⁺¹ cells. V(0) is the mod-p Moore spectrum. V(1) for p ≥ 3, V(2) for p ≥ 5 and V(3) for p ≥ 7 were built by Adams, Smith and Toda by 1971, and nothing new was built for 55 years. Toda, Ravenel and Nave proved that V(n) cannot exist when p is small compared with n, so there is no V(4) at 7 or below, and Ravenel showed that the group that could receive the first differential on v₄ is nonzero at every odd prime, writing that the classical approach could be pushed no further. Hopkins and Smith’s periodicity methods give finite complexes that kill suitable powers of these classes, but not the classes themselves.
Result 308 is two OpenAI papers, 98 pages, both dated September 23, 2026, and made public October 6. The first claims that for every n there is some prime, not named and with no bound given, with a finite spectrum whose Brown–Peterson homology is the quotient by p and v₁ through vₙ, with its canonical comodule structure: Smith–Toda complexes at every height, when the prime may vary. The second, by an independent method, claims that V(4) exists at the prime 1009.
What they don’t claim
One prime for all heights, a bound on the prime, all large primes, or any multiplication on the new complexes. V(4) at 11 and the other primes below 1009 stays open. On the choice of 1009, the paper says only that it makes the finite inequalities in its calculation generous; it is the first prime after 1000.
Where it stands
- Posted as preprints
- OpenAI, two papers, 98 pages, dated September 23, 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. In the first twelve days we found no public specialist comment and no reported error.
How the proof goes
- All primes at once. The varying-primes paper works in an ultraproduct over all odd primes, where a p-fold power operation, made exact by a single Euler class, behaves uniformly in p. It then pulls a finite diagram back to one prime.
- Just in time, at 1009. The second paper rebuilds V(0) through V(3) with unital products, then shows that v₄ survives the Adams–Novikov spectral sequence. Only four possible obstructions remain; the first, a differential of length 2,017, is the one Ravenel’s warning was about, and the paper shows it is zero by an algebraic argument (it does not cite that passage; the link is ours). The last is killed by a relation involving β^((p − 1)²): the product it kills sits at filtration 2,032,133, just before the would-be differential of length 2,034,145.
The numbers in the video
- Ours: at the prime 5, through degree 6,248, the quotient rings shrink from 841,298 dimensions to 3,857, then 82, 7 and 2 as p, v₁, … are killed (algebra only).
- Ours: at 1009, v₄ sits in degree 2,072,977,845,120, and (p − 1)² = 1,016,064.
- Ours: before the release, V(4) was open at every prime from 11 on; at each prime p ≥ 11 the open window ran from height 4 to (p − 1)/2.
Read the source
- Finite Smith–Toda Complexes at Varying Primes (PDF), 54 pages.
- A Finite Smith–Toda Complex V(4) at the Prime 1009 (PDF), 44 pages.
- The collection’s catalog, where this is result 308.
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
In stable homotopy theory, Brown–Peterson homology at a prime p has coefficients a polynomial ring in classes v₁, v₂, …, with vₙ in degree 2(pⁿ − 1), and with v₀ standing for p itself. A Smith–Toda complex V(n) is a finite spectrum whose Brown–Peterson homology is that ring with p and v₁ through vₙ killed, built from 2ⁿ⁺¹ cells. Our computer shows the ring thinning as the classes are killed: at the prime 5, through degree 6,248, the dimension drops from about 841,000 to 3,857, then 82, 7 and 2. V(0) is the mod-p Moore spectrum. V(1) for p ≥ 3, V(2) for p ≥ 5, and V(3) for p ≥ 7 were built by Adams, Smith and Toda by 1971. Then nothing new was built for 55 years. Toda, Ravenel and Nave proved that V(n) cannot exist when p is small compared with n: by Nave, there is no V((p + 1)/2) for p ≥ 7, and with the earlier results, no V(4) at 7 or below. Ravenel also showed that the group that could receive the first differential on v₄ is nonzero at every odd prime, and wrote that the classical approach could be pushed no further. Hopkins and Smith’s periodicity methods give finite complexes that kill suitable powers of these classes, but not the classes themselves. Our computer lays out the known table by prime and height. Before the release, V(4) was open at every prime from 11 on, and at each prime p ≥ 11 the open window ran from height 4 to (p − 1)/2.
1:49The claims
Result 308 is two OpenAI papers, 98 pages, both dated September 23 and made public October 6, 2026. The first claims that for every n there is some prime, not named and with no bound given, with a finite spectrum whose Brown–Peterson homology is the quotient by p and v₁ through vₙ, with its canonical comodule structure: Smith–Toda complexes at every height, when the prime may vary. The second, by an independent method, claims that V(4) exists at the prime 1009. Not claimed: one prime for all heights, a bound on the prime, all large primes, or any multiplication on the new complexes. V(4) at 11 and the other primes below 1009 stays open. On the choice of 1009, the paper says only that it makes the finite inequalities generous; it is the first prime after 1000.
2:47How the proof goes
The varying-primes paper works in an ultraproduct over all odd primes, where a p-fold power operation, made exact by a single Euler class, behaves uniformly in p. It then pulls a finite diagram back to one prime. The second paper rebuilds V(0) through V(3) with unital products, then shows that v₄ survives the Adams–Novikov spectral sequence. Only four possible obstructions remain, and the last is killed by a relation involving β^((p − 1)²). Our computer recomputes the paper’s numbers: (p − 1)² is 1,016,064, and the product it kills sits at filtration 2,032,133, just before the would-be differential of length 2,034,145. Ravenel’s warning was about the first of the four, a differential of length 2,017; the paper shows it is zero by an algebraic argument. It does not cite that passage, and the link is ours. Our computer also finds the scale: at 1009, v₄ sits in degree about 2 trillion.
3:59Where it stands
Both papers are unrefereed and unchanged since release, and there is no Lean formalization. In the first 12 days we found no public specialist comment and no reported error. If they hold, Smith–Toda complexes exist at every height, at some prime for each height, and V(4), the first new height in 55 years, exists at the prime 1009.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.