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Math-2778-A1 · revision r1 · on Math-2778

OpenAI claim summary for Math-2778

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 result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 014; 2026-10-05).

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

Manuscript

Contents
  1. Generic Ramanujan for exceptional groups over function fields
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Generic Ramanujan for exceptional groups over function fields

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-10-05), the asserted result is that globally generic cuspidal representations of split adjoint exceptional groups over function fields are tempered at ramified as well as unramified places, in every characteristic.

Mathematical statement of the claimed result

Manuscript 1.
Let $F$ be a global function field and $G/F$ split connected adjoint absolutely simple of one of the exceptional types $G_2,F_4,E_6,E_7,E_8$. For every complex globally generic cuspidal automorphic representation $\pi=\bigotimes'_v\pi_v$ of $G(\mathbb A_F)$, the claimed conclusion is \[ \pi_v\text{ is tempered for every place }v. \] Ramified places are included. The statement imposes no characteristic, ramification-depth or local-representation-type restriction.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), the companion unramified result supplies global purity. Local functional equations compare semisimple parameters independently of monodromy; regularity of the adjoint local factor then identifies the monodromy orbit needed to apply the local temperedness criterion.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources