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

OpenAI · September 27, 2026

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Abstract

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.

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Cite

@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}
}

Family 318: 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.

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