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Math-2516-A1 · revision r1 · on Math-2516

OpenAI claim summary for Math-2516

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: open. 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.

The fixed-complete-coefficient-ring corollary states exactly O-linear bounded-residue-defect Morita finiteness for every fixed complete characteristic-zero DVR with algebraically closed characteristic-p residue field. Within these assumptions, this is a claimed affirmative answer to the stated question.

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

Manuscript

Contents
  1. Integral block finiteness over a fixed complete valuation ring
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Integral block finiteness over a fixed complete valuation ring

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Bounded defect is claimed to yield finitely many integral Morita classes over Witt vectors, and over any fixed complete valuation coefficient ring of mixed characteristic, provided its residue field is algebraically closed.

Mathematical statement of the claimed result

Result 1.
For $\mathcal O=W(\overline{\mathbb F}_p)$ and integer $M\ge1$, the source claims finitely many $\mathcal O$-linear Morita classes among $\mathcal O[G]b$, as $G$ ranges over finite groups and blocks $b$ have defect-group order at most $M$. Its complete-ring corollary states more generally: fix a complete discretely valued ring $\mathcal R$, whose fraction field has characteristic zero and whose residue field $\ell$ has characteristic $p$ and is algebraically closed. Then \[ \#\{\mathcal R[G]b:\ G\text{ finite},\ b\text{ primitive central idempotent},\ |D_{\bar b}|\le M\}/\mathcal R\text{-linear Morita equivalence}<\infty. \] Only the block and finite group vary; the coefficient ring remains fixed.

Outline of the argument

The proof combines integral bounds for basic orders with a bound on their Morita--Frobenius behavior. Rationality and Cartan controls feed the integral finiteness criterion, and lifting from the residue field handles fixed ramified coefficient rings as well as Witt vectors.

Formal verification

Family 203 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources