Screen a decomposition plan by first trying to prove all of its subgoals directly, then identifying the key stuck points if the plan does not fully go through. Use when a decomposition plan is created.
Find relevant math results, constructions, examples, counterexamples, and background references for a statement. Use when you need context for a new problem, supporting references for constructing examples or counterexamples, or external results while proving subgoals.
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.
Verify a markdown proof in the order it is written. Use when the task is to check local correctness, theorem applicability, and reasoning gaps statement by statement through a paper-style proof.
Launch one sub-agent per decomposition plan after direct screening has identified the key stuck points for each plan. Use when all current plans have been screened by direct proving, none fully solves the problem, and parallel recursive work is needed.
Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail. Use when you are stuck in reasoning and want to see where the assumptions take effect and gain intuition, when a proposed conjecture/claim feels fragile or unproved, or when you want to test whether the assumptions can hold while the claimed conclusion fails.
Generate and analyze simpler examples that satisfy both the assumptions and the conclusion of a theorem statement or subgoal. Use when you are stuck in reasoning and need simpler examples to regain traction, when you need simpler examples that satisfy both assumptions and conclusion, or when you want to see where the assumptions take effect and gain intuition.
Synthesize the common stuck points across failed decomposition plans and recursive sub-agent reports. Use when the current batch of decomposition plans has failed.