OpenAI · 318-04 · 2026-09-25 · Topology

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

preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026

Family 318. Chromatic splitting: counterexamples and filtrations.

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

Disproves strong chromatic splitting at height three for primes $p \geq 5$ , and weak splitting for the derived $p$ -completed sphere at heights $p$ ( $p \geq 5$ ) and $p + 1$ ( $p \geq 7$ ). Nevertheless, for $n \geq 1$ and $p > n + 1$ , the overlap $L_{n - 1} L_{K (n)} S_{p}^{\land}$ 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.

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.

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.