check-referenced-statements

name	check-referenced-statements
description	Validate externally referenced theorems by querying arXiv theorem search first and Codex's built-in web search second. Use when a markdown proof cites statements from external papers.

Check Referenced Statements

Validate every external-paper reference used in the proof.

Input Contract

For each cited external theorem/lemma/definition:

• location where it is used,
• the full referenced statement text.

Procedure

1. Query search_arxiv_theorems using the full referenced statement as query.
2. Inspect returned results and compare theorem text directly to the referenced statement in reasoning.
3. Expand the definitions and terminology appearing in the cited statement using the cited paper's context before deciding whether the theorem applies.
4. Check whether the same words in the current proof mean the same thing as they do in the cited paper. In mathematics, identical words can carry different definitions in different contexts.
5. Accept as matched and applicable only when both are true:
   • the result clearly corresponds to the cited statement,
   • the contextual definitions and hypotheses align with the current problem.
6. If the theorem exists but the current proof uses different definitions, hypotheses, or ambient objects, record a critical error for incorrect application.
7. If no match is found, use Codex's built-in web search with the same statement text.
8. If still not found, emit a critical error:
   • location: where the citation is used,
   • issue: referenced theorem appears non-existent or incorrectly cited.
9. Persist each reference check in reference_checks.

Do not rely on dedicated comparison utility code; perform comparison through careful reasoning.

Output Contract

Append records to reference_checks like:

{
  "location": "Lemma 2",
  "referenced_statement": "Exact statement text",
  "context_expansion": "In the cited paper, 'regular' means regular with respect to the valuation topology.",
  "arxiv_match_found": false,
  "web_match_found": false,
  "critical_error": {
    "location": "Lemma 2",
    "issue": "Referenced external theorem was not found in arXiv search or Codex built-in web search."
  }
}

Tools

• search_arxiv_theorems
• memory_append
• Codex's built-in web search