frenzymath/danus/construct-counterexamples

Construct Counterexamples

Actively falsify proposed conjectures or intermediate claims by finding examples that satisfy the assumptions but violate the claimed conclusion.

Input Contract

Read:

• the specific conjecture/claim to test
• active branch assumptions
• candidate lemmas/proof steps
• current immediate_conclusions and toy_examples
• previously found counterexamples that can be reused against new claims

Procedure

1. Identify the assumptions that must hold and the conclusion to fail.
2. Use reasoning, decomposition, and retrieval to search for standard obstructions, pathological constructions, or previously known counterexamples.
3. Decide status:

• refuted: assumptions hold and the claim fails
• not_refuted: no counterexample found yet
• inconclusive: search space unclear or partially explored

1. If the search produces a concrete example that is informative but is not actually a counterexample, save that example as well in toy_examples.
2. If refuted, store the counterexample for reuse against future claims and mark impacted branches/lemmas as invalid.
3. If no counterexample is found, treat that only as evidence that the claim may be correct, not as a proof.

Output Contract

Publish to global memory with gm_add (kind counterexample): claim = what is refuted/tested, evidence = the candidate construction, plus these fields:

{
  "target_claim": "...",
  "candidate_counterexample": "...",
  "status": "refuted|not_refuted|inconclusive",
  "assumptions_satisfied": ["..."],
  "failed_conclusion": "...",
  "impact": "...",
  "branch_id": "optional",
  "subgoal_id": "optional"
}

If status="refuted" and it kills a branch, also publish a dead_end finding (gm_add, kind dead_end) so siblings skip that branch.

If the search produced a concrete non-refuting example, also publish an example finding (gm_add, kind example):

{
  "example": "...",
  "why_relevant": "constructed while testing the claim ...",
  "assumptions_satisfied": ["..."],
  "conclusion_verified": true,
  "where_assumptions_take_effect": "...",
  "observed_pattern": "...",
  "supports_branch_ids": ["optional"],
  "subgoal_id": "optional"
}

Do this whenever the constructed example is useful enough to test future claims or clarify the current branch, even if it did not refute the target claim.

Tools

• gm_add (publish counterexample / dead_end / example findings)
• gm_search (recall stored counterexamples to reuse against new claims)
• Codex built-in web search and search_arxiv_theorems to find standard counterexample patterns

Failure Logging

If no meaningful counterexample space is identified, append:

• events.event_type="counterexample_space_unclear"