Does it split?

The claims, in plain words

At a prime p, stable homotopy theory sorts the sphere into chromatic heights, 0, 1, 2 and up. To rebuild everything up to height n, you glue height n onto the heights below it, along an overlap: the part of the height-n world that the lower heights can see. Around 1993 Hopkins conjectured that this overlap is as simple as it could be.

In the strong form, the overlap is a wedge of 2ⁿ shifted copies of lower-height spheres, one for each subset of the numbers 1 to n. Our computer lists them at height 3: 8 pieces, shifted by 0, −1, −3, −4, −5, −6, −8 and −9. In the weak form, at least the first piece, the lower sphere itself, splits off.

Height 1 can be done by hand, and height 2 was known at primes 5 and up and at 3. At the prime 2, Beaudry showed in 2017 that the strong form fails, and Beaudry, Goerss and Henn later showed the weak form still holds. In 2024 Barthel, Schlank, Stapleton and Weinstein showed the rational shadow is right at every height. At height 3 and above, the integral question was open, apart from one partial result posted this August.

Result 318 is five OpenAI papers, 372 pages, dated September 25 to 27 and made public October 6, 2026. They give five answers. First, for primes bigger than n + 1, the overlap is built from exactly the predicted pieces, glued in order in a filtration; a second paper does height 3 explicitly. Third, at height 3 and primes 5 and up, that gluing cannot be fully undone: the strong wedge form fails.

Fourth, at height 3 and the prime 3, the overlap cannot be built from lower spheres in finitely many steps at all. Fifth, the weak form, for the p-completed sphere, fails at heights p and p + 1, for primes 5 and up and 7 and up: the lower sphere does not even split off. Our computer maps the claims onto heights and primes, and the filtration and the weak failures never meet. Not claimed: the weak form at height 3, or at primes large compared with the height, where several authors expect the conjecture to hold, or anything at the prime 2.

Where it stands

Posted as preprints

OpenAI, five papers, 372 pages, dated September 25–27, 2026

Peer reviewed

Not yet

Checked in Lean

No formalization for this item

Last checked October 19, 2026. All five papers are unrefereed and unchanged since release, with no Lean formalization, and four of them cite that six-week-old human preprint, two as a direct input. Its revised version has a footnote: on September 15, Agnès Beaudry and Tomer Schlank announced an AI-generated disproof of the weak chromatic splitting conjecture for primes which are small relative to the height. We could not find that announcement, so whether it is the release’s weak-splitting paper is unverified. In the first 12 days we found no specialist comment on these papers and no reported error. OSVerify has not reviewed them.

How the proofs go

1. A rational test at height 3. The filtration and the strong counterexample rest on p-adic geometry, Lubin–Tate and Drinfeld towers and the Fargues–Fontaine curve, building on Barthel, Schlank, Stapleton and Weinstein; the counterexample also builds on a human preprint from August 2026. The strong counterexample is a rational test: at height 3, a canonical map is nonzero in degree −3, where the wedge would force zero. Our computer’s bookkeeping shows there is room for such a test first at height 3, and it reproduces the Lie algebra ranks, 0, 8, 20 and 35, that the paper cites from a support script not included in the release.
2. One class that cannot survive. The weak counterexample is short and classical. It follows one class, β₁ to the power (p − 1)². At p = 5 that is β₁¹⁶, in stem 608. The class is alive one height down, but a vanishing line kills it at height p, so the lower sphere cannot split off. Our computer checks where the prime bounds 5 and 7 come from.

The numbers in the video

• Ours: at height 3, the eight predicted pieces, shifted by 0, −1, −3, −4, −5, −6, −8 and −9.
• Ours: in degree −3 at height 3, the wedge would force a canonical map to be zero where the source and the expected target each have dimension one; the first height with room for such a test is 3.
• Ours: the Lie-algebra ranks 0, 8, 20, 35 that one paper cites from a support script not included in the release.
• Ours: the weak counterexample’s witness is β₁¹⁶ in stem 608 at p = 5; the prime bound fails at p = 3, works at height 5 only for p = 5, and at heights p and p + 1 for p ≥ 7.
• Ours: on a grid of heights and primes, the filtration and the weak failures never meet.

Read the sources

• Filtered Chromatic Splitting at Generic Primes (PDF), 110 pages.
• The Height-Three Chromatic Overlap: An Explicit Filtration and Its Attachments (PDF), 112 pages.
• A Rational Obstruction to Strong Chromatic Splitting at Height Three (PDF), 49 pages.
• Failure of Finite Assembly for a Chromatic Overlap at the Prime Three (PDF), 72 pages.
• Counterexamples to Weak Chromatic Splitting: Sphere Kernels and Descent Exponents (PDF), 29 pages.
• The collection’s catalog, where this is result 318.

The 5 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.

• Filtered chromatic splitting at generic primes main paper
• The height-three chromatic overlap: an explicit filtration and its attachments companion
• A rational obstruction to strong chromatic splitting at height three companion
• Failure of finite assembly for a chromatic overlap at the prime three companion
• Counterexamples to weak chromatic splitting: sphere kernels and descent exponents companion

Transcript

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.