OpenAI · 319-01 · 2026-09-26 · Topology

Counterexamples to the Hahn-Wilson conjecture at height two

preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026

Family 319. Counterexamples to the Hahn–Wilson conjecture at height two.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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 $BP ⟨ 2 ⟩$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

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 $BP ⟨ 2 ⟩_{p}^{\land}$ . The same spectrum satisfies both localization comparisons $L_{2}^{f} X ≃ L_{2} X$ and $L_{T (2)} X ≃ L_{K (2)} X$ .

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.