Remark: Family 319. Counterexamples to the Hahn–Wilson conjecture at 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 $\mathrm{BP}⟨2⟩$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.