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Math-3023 · mathematics · Topology

Finite assembly of the height-three chromatic overlap

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
At $p=3$, can $L_{2}L_{K(3)}S_{3}^\wedge $ be built from $L_{0}S_{3}^\wedge $, $L_{1}S_{3}^\wedge $ and $L_{2}S_{3}^\wedge $ by finitely many shifts, sums, cofibers and retracts in the category of E(2)-local modules over the derived three-complete sphere?

Short English statement

At $p=3$, can $L_{2}L_{K(3)}S_{3}^\wedge $ be built from $L_{0}S_{3}^\wedge $, $L_{1}S_{3}^\wedge $ and $L_{2}S_{3}^\wedge $ by finitely many shifts, sums, cofibers and retracts in the category of E(2)-local modules over the derived three-complete sphere?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3023-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3023

Manuscript 1 (family 318, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Finite assembly of the height-three chromatic overlap

OpenAI's claim

At prime three and height three, the overlap is claimed to lie outside the ordinary thick subcategory generated by lower local spheres, even allowing arbitrary finite cofibers and retracts of derived-complete sphere modules. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
At prime $3$, with completed sphere $S=S_3^\wedge$, OpenAI claims \[ L_2L_{K(3)}S\notin\operatorname{Thick}\{L_0S,L_1S,L_2S\} \] in the category of $E(2)$-local $S$-modules. Here $L_j$ is $E(j)$-localization, $L_{K(3)}$ is Morava-$K(3)$ localization, and the thick subcategory allows finite cofibers, suspensions and retracts. Thus arbitrary finite attachments, including Moore corrections, cannot assemble this chromatic overlap from the indicated lower-height localized spheres.

Outline of the argument

Manuscript 1 (family 318, 2026-09-25): Integral connected and etale markings retain an exceptional norm-character line on the height-two stratum. Full principal-unit action separates that line from every periodicity power carried by the permitted inputs. Comparisons through ordinary finite-stage coefficients and actual homotopy preserve the character across the whole deformation, giving a representation obstruction to any finite assembly.

Formal verification

Family 318 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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