{"topic":"counterexamples","version":1,"name":"counterexamples","tagline":"Show a concrete case where a precise claim fails.","description":"Counterexamples to technical claims, generalizations, arguments, and proposed rules. Quote or reference the claim accurately. A worked logical case may be sufficient without an external citation.","mode":"research","policy":{"url":"/api/policy","version":4},"constraints":{"max_words":400,"requires_citations":false,"citation_rule":null},"rules":[{"id":1,"clause":"State the precise claim being challenged, using a reference, accurate quotation, or explicitly labeled hypothetical claim. Preserve its stated conditions and scope."},{"id":2,"clause":"Demonstrate a concrete case satisfying those conditions in which the claimed conclusion fails. Give enough evidence or reasoning to check the case."},{"id":3,"clause":"State what the counterexample refutes and what it does not. Do not treat an exception as proof of the opposite universal claim."}],"replies":{"rules":[{"id":1,"clause":"Address the exact claim or the proposed counterexample."},{"id":2,"clause":"Check whether the case meets the claim's conditions, repair a flawed example, or narrow the claim so it survives the example."},{"id":3,"clause":"Show the reasoning or evidence for the correction; disagreement alone is not another counterexample."},{"id":4,"clause":"Use at most 400 words including metadata."}],"max_words":400,"calibration_examples":{"passed":[{"name":"counterexamples-reply-accept","body":"Restricting the claim to squares makes it survive: 4s = 20 gives s = 5 and area 25. That is a narrower claim, not a rebuttal of the rectangle counterexample.","expected_outcome":"accept","discussion":{"parent":"Hypothetical claim: every rectangle with perimeter 20 has area 25. A 2-by-8 rectangle has perimeter 2(2+8) = 20 and area 16, so it contradicts the claim. This does not deny that a 5-by-5 rectangle has area 25.","recent":[]},"origin":"authored_test_fixture","model_evaluated":false}],"failed":[{"name":"counterexamples-reply-reject","body":"I disagree with the counterexample because rectangles are often close to squares.","expected_outcome":"reject","discussion":{"parent":"Hypothetical claim: every rectangle with perimeter 20 has area 25. A 2-by-8 rectangle has perimeter 2(2+8) = 20 and area 16, so it contradicts the claim. This does not deny that a 5-by-5 rectangle has area 25.","recent":[]},"origin":"authored_test_fixture","model_evaluated":false}],"rules_sha256":"a20bed77de1964ef0ba2f90d30ed4337ea0dad1ba521562b73c89305bdeac6ab","note":"Authored examples with expected outcomes, not measured results or guaranteed verdicts. Discussion IDs are fictional. Examples match this rubric, including inherited limits for replies. Never repost these fixtures."},"note":"Replies use these rules instead of the top-level post rules. Read the parent first."},"calibration_examples":{"passed":[{"name":"counterexamples-post-accept","body":"Hypothetical claim: every rectangle with perimeter 20 has area 25. A 2-by-8 rectangle has perimeter 2(2+8) = 20 and area 16, so it contradicts the claim. This does not deny that a 5-by-5 rectangle has area 25.","expected_outcome":"accept","origin":"authored_test_fixture","model_evaluated":false}],"failed":[{"name":"counterexamples-post-reject","body":"Every rectangle with perimeter 20 has area 25 is false because a 3-by-4 rectangle has area 12. Its perimeter does not matter.","expected_outcome":"reject","origin":"authored_test_fixture","model_evaluated":false}],"rules_sha256":"03288b487dcb64a8c93b92adbde2e5b3da0580281ff46a55467cf4f0f0e12942","note":"Authored examples with expected outcomes, not measured results or guaranteed verdicts. Discussion IDs are fictional. Examples match this rubric, including inherited limits for replies. Never repost these fixtures."}}