Result 319, Topology

Counterexamples to finite generation at chromatic height two

Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed BP⟨2⟩\mathrm{BP}\langle2\rangle by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

Disproof or counterexample

The bigger picture

Why it matters

In topology, passing two important comparison tests does not necessarily mean an object can be assembled from a standard building block. This manuscript reports counterexamples that separate these two ways of recognizing structure.

What changes?

The unreviewed manuscript reports counterexamples to the Hahn-Wilson conjecture at chromatic height two. For every sufficiently large prime p, it constructs a spectrum, an object encoding stable topological information, that is connective, meaning it has no negative-degree homotopy, and p-complete, retaining information completed at p. Its finiteness classification is exact fp-type two. Yet it cannot be built from the specified standard form of completed BP⟨2⟩ using finitely many sums, degree shifts, mapping cones and retracts, which select direct summands.

What does that help mathematicians do?

The same examples satisfy both stated comparisons: finite and ordinary height-two localization agree, and telescope and Morava K-theory localization agree at height two. These operations isolate particular layers of topological information. Consequently, even both agreements together do not guarantee finite construction from the specified generator, under the stated assumptions. Researchers can therefore rule out these comparison properties as sufficient criteria for membership in that generator's ordinary thick subcategory.

Are there practical applications?

The immediate value is foundational: the examples sharpen the distinction between what localization detects and what finite assembly can produce. They give researchers a concrete obstruction to proposed building-block classifications at height two. Any revised criterion must account for this distinction; the reported result does not extend the counterexamples to every prime or every chromatic height.

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.

Manuscript

Counterexamples to the Hahn-Wilson conjecture at height two

September 26, 2026 79 pages

We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of BP⟨2⟩p∧\mathrm{BP}\langle2\rangle_p^\wedge. The same spectrum satisfies both localization comparisons L2fX≃L2XL_2^fX\simeq L_2X and LT(2)X≃LK(2)XL_{T(2)}X\simeq L_{K(2)}X.

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Cite (BibTeX) 
@misc{OAI:Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026,
  author = {{OpenAI}},
  title = {{Counterexamples to the Hahn--Wilson conjecture at height two}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf}{OAI:Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.