Result 318, Topology

Chromatic splitting: filtrations and counterexamples

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and $p+1$ (p ≥ 7). Nevertheless, for n ≥ 1 and $p\gt n+1$ , the overlap $L_{n-1}L_{K(n)}S_p^\wedge$ admits a $2^n$ -stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of $E(2)$ -local modules over the derived completed sphere.

Disproof or counterexample

The bigger picture

Why it matters

Chromatic topology studies the sphere spectrum, a basic building block of stable topology, through layers indexed by primes and heights. These manuscripts claim that expected pieces can fit together without separating into independent parts.

What changes?

For the derived p-completed sphere, every n at least 1 and prime p greater than n+1 reportedly gives the overlap of height-n and lower-height localizations a filtration with 2^n stages, built successively from predicted localized-sphere pieces. Nevertheless, strong splitting fails at height three for primes at least 5. The canonical weak splitting map has no homotopy retraction at height p for primes p at least 5, or height p+1 for primes p at least 7.

What does that help mathematicians do?

A filtration records how pieces attach, rather than declaring them independent. At height three and primes at least 5, the reported nonzero first height-one attachment makes that distinction concrete. At prime three, the obstruction is stronger: no finite sums, shifts, cofibers or retracts of rational, height-one and height-two local spheres build the overlap within E(2)-local modules over the derived 3-completed sphere. This rules out finite assembly in that specific category.

Are there practical applications?

The immediate value is foundational: researchers gain a layered replacement for a failed decomposition, alongside limits on where such constructions work. The map from the general filtration's first stage to the overlap is the canonical localization map, preserving that specified relationship. This supplies structured attachment problems for further study, not a practical algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

5 manuscripts

Filtered chromatic splitting at generic primes

September 25, 2026 110 pages

For every n ≥ 1 and prime $p\gt n+1$ , we construct a $2^n$ -stage ordered filtration of $L_{n-1}L_{K(n)}S_p^\wedge$ with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Filtered-chromatic-splitting-at-generic-primes-September-25-2026,
  author = {{OpenAI}},
  title = {{Filtered chromatic splitting at generic primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf}{OAI:Filtered-chromatic-splitting-at-generic-primes-September-25-2026}},
  year = {2026}
}

The height-three chromatic overlap: an explicit filtration and its attachments

September 27, 2026 112 pages

For every prime p ≥ 5, we construct an explicit eight-stage filtration of $L_2L_{K(3)}\mathbb S_p^\wedge$ by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026,
  author = {{OpenAI}},
  title = {{The height-three chromatic overlap: an explicit filtration and its attachments}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf}{OAI:The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026}},
  year = {2026}
}

A rational obstruction to strong chromatic splitting at height three

September 25, 2026 49 pages

For every prime p ≥ 5, the canonical map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ for the sphere spectrum S is nonzero on π₋₃. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying $E(2)$ -local spectra without specified summand maps.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026,
  author = {{OpenAI}},
  title = {{A rational obstruction to strong chromatic splitting at height three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf}{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026}},
  year = {2026}
}

Failure of finite assembly for a chromatic overlap at the prime three

September 25, 2026 72 pages

At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of $E(2)$ -local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026,
  author = {{OpenAI}},
  title = {{Failure of finite assembly for a chromatic overlap at the prime three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf}{OAI:Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026}},
  year = {2026}
}

Counterexamples to weak chromatic splitting: sphere kernels and descent exponents

September 27, 2026 29 pages

For the derived p-completion of the sphere, the canonical weak chromatic 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 classical sphere product $\beta_1^{(p-1)^2}$ gives a nonzero kernel class in both ranges.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026,
  author = {{OpenAI}},
  title = {{Counterexamples to weak chromatic splitting: sphere kernels and descent exponents}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf}{OAI:Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026}},
  year = {2026}
}