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Math-2261-A1 · revision r1 · on Math-2261

OpenAI claim summary for Math-2261

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 313, 2026-09-24): Theorem 1.1 is exactly integer-degree finite generation over the p-adic integers for every prime and positive height, and explicitly excludes the different Picard-graded extension mentioned in the background.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Morava K- and E-theory 1 — Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree.
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Morava K- and E-theory 1 — Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree.

OpenAI's claim

At every prime and positive height, each integer-degree homotopy group of the K(n)-local sphere is claimed to be finitely generated over Z_p, equivalently for every localized finite p-local spectrum. 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 every prime $p$, positive height $n$ and integer degree $t$, OpenAI claims \[ \pi_tL_{K(n)}S_p^\wedge\text{ is finitely generated over }\mathbb Z_p. \] Equivalently it claims the same for $\pi_tL_{K(n)}F$ for every finite $p$-local spectrum $F$. Here $K(n)$ is Morava $K$-theory, $L_{K(n)}$ its localization and $\mathbb Z_p$ the $p$-adic integers. Negative as well as nonnegative degrees are included.

Outline of the argument

Manuscript 1 (family 313, 2026-09-24): Finite division-point markings and compact-support geometry control stabilized transfer images in continuous Morava coefficient cohomology. Laurent-series models and Frobenius contraction remove a positive-order acyclic kernel, while Koszul extension arguments promote stable-image finiteness to the required coefficient groups. Sphere descent completes the module-finiteness conclusion, and finite cofiber sequences and retracts extend it to finite spectra.

Formal verification

Family 313 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