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The chromatic splitting conjecture

From OpenAI Aftermath

The chromatic splitting conjecture, proposed by Hopkins and recorded by Hovey, describes the overlap Ln−1LK(n)S between the K(n)-local sphere and the lower chromatic heights. Its strong form predicts a wedge decomposition into shifted lower local spheres; its weak form asks only that the canonical unit Ln−1S → Ln−1LK(n)S be split. Five manuscripts dated 25 and 27 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim a filtered form of the predicted decomposition for primes p > n + 1, a disproof of the strong form at height three for p ≥ 5, a failure of finite assembly at p = n = 3, and counterexamples to the weak form for the p-complete sphere. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

  • As recounted in the introductions of the manuscripts:
    • Hopkins' conjecture, recorded by Hovey (Conjecture 4.2), proposes classes, factorizations and summands that would split a canonical cofiber sequence; the weak consequence asks for a retraction of the canonical unit. Barthel and Beaudry review a revised strong formulation (Conjecture 6.3 and Remark 6.4), which keeps the same pattern of lower local spheres at height three for p ≥ 5.
    • Hovey's 1993 account records the height-one case and the height-two case for p > 3 as known, attributing the latter to Hopkins using the calculations of Shimomura and Yabe. Goerss, Henn and Mahowald proved the height-two splitting at p = 3.
    • At height two and the prime 2, Beaudry disproved the original strong formula, and Beaudry, Goerss and Henn established a corrected decomposition with additional Moore-spectrum terms that keeps the unit as a retract. Their Conjectures 1.1.10 and 1.1.11 separate the strong formulation from the weak formulation for completed finite inputs.
    • Barthel, Schlank, Stapleton and Weinstein computed the rational homotopy of the K(n)-local sphere at every prime and height as an exterior algebra on generators in degrees 1 − 2i, 1 ≤ i ≤ n. At height three the degrees are 0, −1, −3, −4, −5, −6, −8 and −9, which match the summands of the predicted wedge, so rational degrees alone do not decide the strong form.

The original results

  • Filtered splitting at generic primes (25 September 2026). Theorem 1.1 of the first manuscript states that for every n ≥ 1 and every prime p > n + 1 the overlap Ln−1LK(n)S of the p-complete sphere has an ordered filtration with 2n stages whose successive cofibers are the classical splitting pieces, a shift of Ln−max IS for each nonempty subset I of {1, …, n} together with Ln−1S, and whose first stage maps to the overlap by the canonical unit. It is deduced from classes in the K(n)-local sphere whose products give K(t)-local bases at every lower height t. The manuscript states that the filtration keeps the attaching maps and asserts neither the wedge decomposition nor a retraction of the unit, and it makes no sharpness claim for the prime bound.
  • An explicit filtration at height three (27 September 2026). The second manuscript constructs, for every p ≥ 5, an explicit eight-stage filtration of L2LK(3)S with cofibers L2S, Σ−1L2S, Σ−3L1S, Σ−4L1S and four rational pieces in degrees −5, −6, −8 and −9, and identifies all its attaching maps by two signed fracture formulas relative to chosen local maps. It states that the construction does not depend on the first manuscript, and that, by the theorem of the third manuscript, the first height-one attaching map is nonzero.
  • A rational obstruction to strong splitting (25 September 2026). Theorem 1.1 of the third manuscript states that for every prime p ≥ 5 the canonical map L0LK(3)S → L0LK(2)LK(3)S is nonzero on π−3. Corollary 1.2 concludes that L2LK(3)S is not equivalent to the predicted wedge, so that the height-three strong formula is false in this range even as an equivalence of underlying E(2)-local spectra without specified summand maps.
  • Failure of finite assembly at the prime three (25 September 2026). Theorem 1.1 of the fourth manuscript states that at p = n = 3 the overlap L2LK(3)S does not lie in the thick subcategory generated by L0S, L1S and L2S in E(2)-local modules over the 3-complete sphere, so that it cannot be built from the lower local spheres by finitely many sums, shifts, cofibers and retracts. The manuscript states that this answers the finite-assembly question negatively at this prime and height only, and that it does not by itself address whether the unit retracts there.
  • Counterexamples to weak splitting (27 September 2026). Theorems 1.1 and 1.2 of the fifth manuscript state that for the derived p-completion of the sphere the canonical weak splitting map has no homotopy retraction at height p for every prime p ≥ 5 and at height p + 1 for every prime p ≥ 7. The witness is the classical product β1(p−1)2, whose image at height p − 1 is nonzero and lies in the kernel of the unit; at p = 5 it is β116 in degree 608. Theorem 1.4 gives a second argument at height p that compares descent exponents for a finite subgroup of the Morava stabilizer group.
    • Method, as described: the filtration manuscripts represent stabilizer cohomology classes by sphere classes and test them at every lower height through coefficients on the characteristic-p deformation strata of Lubin–Tate space. The rational obstruction detects the degree-three trace class tr(U[V,W]) of the stabilizer and compares its height-three and height-two realizations through modifications of Fargues–Fontaine bundles, presented both by projective space and by Drinfeld space. The finite-assembly manuscript starts from the coheight-one work of Barthel, Mann, Ray, Schlank, Senger, Weinstein and Zhou on the chromatic splitting conjecture, cited as a preprint dated 26 August 2026, and from earlier work of Gross and Torii. The weak-splitting counterexamples use the Hopkins–Miller calculation recorded by Heard for detection at height p − 1 and a vanishing line at the upper height.
  • Sources. Commit 1581d4c of the openai/math repository, folders Filtered chromatic splitting at generic primes, The height-three chromatic overlap: an explicit filtration and its attachments, A rational obstruction to strong chromatic splitting at height three, Failure of finite assembly for a chromatic overlap at the prime three and Counterexamples to weak chromatic splitting: sphere kernels and descent exponents.
  • Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date Work Relation Public claim Public status (checked 17 October 2026)
2026-10-07 Agnes Beaudry and Tomer Schlank, Hexagon 2610.00138v1, titled Weak chromatic splitting fails at n = p for every odd prime p Parallel work The note proves that the weak chromatic splitting conjecture fails for (n, p) = (3, 3) and for every pair with p ≤ n < √2(p − 1), in particular at n = p for every odd prime p, because the unit map is not injective on homotopy groups. Its abstract states that the disproof "was produced autonomously by Claude" in a session on Freyd's generating hypothesis, and that the note is a human write-up to which the authors "did not contribute any ideas". Its acknowledgments state that the transcript was shared with the authors on 8 September 2026 and that the result was announced at the 2026 Clay Research Workshop on Telescopic Homotopy Theory and Algebraic K-theory on 21 September 2026. The note uses the classes β1(p−1)2 for p ≥ 5 and β15 for p = 3. The pairs treated in the fifth manuscript, height p for p ≥ 5 and height p + 1 for p ≥ 7, satisfy this inequality. Hexagon version 1, CC BY 4.0; the note does not refer to the OpenAI manuscripts.
2026-10-16 J. Jardine, Lattice24, referee reviews rank_120, rank_125, rank_211, rank_202 and rank_289 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) Review (AI-assisted audit) The five reviews, in the order of the manuscripts above, report no gap. The review of the first manuscript notes as a scope limit that its main coefficient computation rests on about 5,000 lines of p-adic geometry whose internal proofs were not checked. The review of the second notes that the nonvanishing of the first height-one attaching map depends on the third manuscript. The reviews of the third and fourth describe partial passes; the review of the fourth states that its result rests on a cohomology computation in Sections 6 and 7 that the review did not verify. The review of the fifth asks for a specialist to confirm the use of a lemma of Heard. Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

  • The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
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