{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0001","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b64c5e95d3723bdf5d330750cad8e7b63fda43c1dd442b15b9883963c2e8acdb","family_id":"qf-density17-counter-full-failure-min","seed":202610177001,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":0,\"truth\":[]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([], [], Yes, does-not-exist)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, does-not-exist)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0002","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f820092847699a94b23929f813ce31819ec58c81e71a1e9a127fc3dce980360b","family_id":"qf-density17-counter-full-failure-min","seed":202610177002,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[true,true,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0003","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cfd32c4867380328ee419cf3f14180c484190d4f37ee1230424be72729ad7a7","family_id":"qf-density17-counter-full-failure-min","seed":202610177003,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[false,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0004","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6026b6796f8bca9d7f17c7ea283f2baab4fe00cbb2ee4838cad009da31c8d387","family_id":"qf-density17-counter-full-failure-min","seed":202610177004,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0005","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bfcc19b3ba47f5039d259e4065d2d693e2403dcc164dde1a5d9467e62b04bde","family_id":"qf-density17-counter-full-failure-min","seed":202610177005,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0006","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb00c33ef4c8488c8aa17b844e641998484726d9ef40dfdc611d793b97e9f330","family_id":"qf-density17-counter-full-failure-min","seed":202610177006,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0007","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"823d12dccbe6c2a2a5a378574dad907c98648d89812a40a053682df8b8c5925d","family_id":"qf-density17-counter-full-failure-min","seed":202610177007,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0008","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39e7f9e36d90f35cb11bc3bb831c7bb782062f0914ecd15726797c6439a583cb","family_id":"qf-density17-counter-full-failure-min","seed":202610177008,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0009","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4888b71222b1808e9b48417042d52107f45414649c039272c8f5bc58a0c18d37","family_id":"qf-density17-counter-full-failure-min","seed":202610177009,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0010","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"144a76b0fbf9bd3f2366363b0adde2c579df17fe061b407c5d932c373eec2134","family_id":"qf-density17-counter-full-failure-min","seed":202610177010,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,true,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0011","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b06fc7ed13a43a56efe30d214db5bf669e35208895edf7cfc53983d767f6ac8","family_id":"qf-density17-counter-full-failure-min","seed":202610177011,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,true,false,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0012","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ce5207eb9c36656b2489367e7543277cf9102aa4214d4af7e5f4f4edefdefbe","family_id":"qf-density17-counter-full-failure-min","seed":202610177012,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0013","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a63b8446aeaefb69d23141cbb56bf0b1d8e67d0d7015cb094a3f88f322d9f25","family_id":"qf-density17-counter-full-failure-min","seed":202610177013,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0014","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08fd2583fa65112e43726dda616c66f8d8da1051c8894a3244625623338144f2","family_id":"qf-density17-counter-full-failure-min","seed":202610177014,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0015","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f36469863e90d9723357fc6c972e936c26f00bebfcf60bb6b63efd481a7f4497","family_id":"qf-density17-counter-full-failure-min","seed":202610177015,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0016","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54237c1d9f34232071e8c16a8f338cc861a98ad32a027f4f77b816ca9cb443e7","family_id":"qf-density17-counter-full-failure-min","seed":202610177016,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0017","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d84cb020c85b554d898dd74fba8ba1306c3d105de87fb1e1f6d8e83acf840a6a","family_id":"qf-density17-counter-full-failure-min","seed":202610177017,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0018","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79f00af6fc78a5384617fb9f9796826fc8131a8c1c1d1eadcce8545eef3d9c5d","family_id":"qf-density17-counter-full-failure-min","seed":202610177018,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,true,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0019","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5be882e210746cd04891033287aa9ca08d996a31a604fd4a03dbe16380c394c","family_id":"qf-density17-counter-full-failure-min","seed":202610177019,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,true,true,false]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0020","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0d0f26cbfdbeaee77acfb119b055a5b2bf1a6dbbe8caee8dde5ed14a156162a","family_id":"qf-density17-counter-full-failure-min","seed":202610177020,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0021","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7dba78e28dbf3e8591b4029a203a34e6c30b8fc40e15a495870535b632e55b2","family_id":"qf-density17-counter-full-failure-min","seed":202610177021,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0022","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"449936e5e81c08b393a817e6ba41b914b557b69857f3690897c43b03c848c0ab","family_id":"qf-density17-counter-full-failure-min","seed":202610177022,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0023","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50046891f2fd6820c7d440110fc021d03920b7ec4b985922dc180b7132a7ffb9","family_id":"qf-density17-counter-full-failure-min","seed":202610177023,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0024","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90230363956f4ac3711a2a218db6432feff9ecc4996ada68bd907b41314a3473","family_id":"qf-density17-counter-full-failure-min","seed":202610177024,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0025","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3440a20de318d2fe9f01d7afb65015eb2dddd99137a6d76637b04e5db545625d","family_id":"qf-density17-counter-full-failure-min","seed":202610177025,"semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,false,true]}"},"fields":{"title":"Complete Failure Set and Least Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).","answer":{"kind":"expression","value":"([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1)","explanation":"Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1)."},"skills":["Complete Failure Set and Least Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0026","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d75753b60a2f55c3e248cea710bc004aa6e24ffdd273a8f136be84403b92d3b","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177026,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":0,\"queries\":[\"untyped\",\"untyped\",\"untyped\"],\"truth\":[]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0027","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e20c8e2137505125893dbe8d852711c09f89da23d559607c513f56a6aa8199c","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177027,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0028","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f42be2c6e11d3e62449159630def4208fe4eb7697bf6a9725b57a793560971a9","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177028,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0029","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"912e2d510065cde96f179e0621d4af09e2663e0aa0476ba0f9d11dc368845375","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177029,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0030","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7d20c5a3c1586a64e587fc725644611dbc6e98ddccf4b654c6b4bd393b474e3","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177030,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0031","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7943652a9b5f73b90e79360c61631e124988f047e8cf225446eb79935dc6cbc","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177031,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0032","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15940c3e37654b980b209a00ffd4565d6fdf5ff74eb8e2ec176da3170f0c07f","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177032,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0033","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29d31e00b9f876dc0ff82954a74f22584bd1771194d5e74ff87321824bcd5caa","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177033,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",1,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0034","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22838431ea1eb71cd09710b8d5f4be355df96c15149e05e7e3ef02a6a126c15a","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177034,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,1,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0035","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76ba99a95ae4326d5531b2654f0d41d659f1752cfd5ba5346aeefbbe5a493ab8","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177035,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,1,\"untyped\"],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0036","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04422ff5dd6d3f03c8fe67501eabd380cfc765637e026f49075bd29dd1e6b148","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177036,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0037","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6d768c48c08cc9b4aee4a97780f311592e8ea0e237e859512842d27877830eb","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177037,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0038","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f735441a11efae7869030e2af2f5d773f1cb166f21793fe21980220079ce0673","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177038,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0039","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6563192febb59f1a78ed9a3821ffccf9971160b8281b85616cf0fff512ff23b","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177039,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0040","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efa13bae857b9f3ea901054266a7c01926b799f23b9e498e76634639b183b7ca","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177040,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0041","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a7a6cb51165f1402c7779afa183fdfc4e38ed38adbafc87a799a95f4879a84f","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177041,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0042","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e3a3549fc061870c37b34e73397e4e19908b721644563f000256598fd22164","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177042,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",1,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0043","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ac249d007bed30cad2c1044a934050ec1a21e22edca7987f8cce69988a04da4","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177043,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,1,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0044","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88f753954006e33506342f0a652d0f8529ec0263669ad67768aa3b6589275607","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177044,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,1,0],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0045","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ba539becfbd49d5d39b9d74d621a91e62d9f7ece1f4eac4c98a35cd4f2dd00","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177045,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0046","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15ed481f4d0bad8a555d25f4fcbc289c3c738c5fc1d89c2367364e3e0e207ef5","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177046,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0047","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a04e496245f5807b6dd65a2a95130ad2bf1bfe7b9c5adce6b25a4dab49863da","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177047,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0048","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd83adf96f3482cceb4a46f06a750d2bb560a60f81cbaab51b1dcbcf996db098","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177048,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0049","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eefdef9a487fbbb84213620242e38a4941e1097c96fea5f17705442660350a5","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177049,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0050","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61f055d48506edd51c7ca29aa9e4beba358b94e203879b96a17d81014b7183b0","family_id":"qf-density17-counter-three-candidate-guards","seed":202610177050,"semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,1],\"truth\":[true,false]}"},"fields":{"title":"Three Domain Guards before Counterexample Evaluation","stem":"The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).","answer":{"kind":"expression","value":"([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes])","explanation":"Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes])."},"skills":["Three Domain Guards before Counterexample Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0051","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be3b0cbbc6652c6c2cfa563b04bd5a5e7c316456dd2fa10eaf4f58b43c7c6425","family_id":"qf-density17-counter-restriction-witness","seed":202610177051,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":0,\"subset\":[],\"truth\":[],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([], [], [], No, No, Yes)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], No, No, Yes)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0052","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d407cee0f2d3b4d70c9a06797bd9b56ce5e8f6578866f155ff36cf834fbd571a","family_id":"qf-density17-counter-restriction-witness","seed":202610177052,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[],\"truth\":[false,true,false],\"witness\":0}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2], [], [], Yes, No, Yes)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], Yes, No, Yes)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0053","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5014c5cd082dac4204bfa351645c45e874b48619902afc6cdadb0e8108f27b59","family_id":"qf-density17-counter-restriction-witness","seed":202610177053,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[0],\"truth\":[false,true,false],\"witness\":0}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2], [0], [0], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0], [0], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0054","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f76a3712171611ef982771501f83a8d8e2a696c2aab049d2aa33a4c8b89ba0c2","family_id":"qf-density17-counter-restriction-witness","seed":202610177054,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[1],\"truth\":[false,true,false],\"witness\":2}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[1], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2], [1], [], Yes, No, Yes)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [1], [], Yes, No, Yes)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0055","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf6220e2436518132aa217accabed1fa49a1c9b147b2c035a85b95363fe2834b","family_id":"qf-density17-counter-restriction-witness","seed":202610177055,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[0,2],\"truth\":[true,true,true],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([], [0, 2], [], No, No, Yes)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 2], [], No, No, Yes)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0056","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12150bbf8f9f14ba9b16cca627ee4c41f76420c6cf1f741d5c7ce740a2ddb7fe","family_id":"qf-density17-counter-restriction-witness","seed":202610177056,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,false,false],\"witness\":4}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0057","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"630cf69faa370033370db0c405f561ab4264618136ca1fe2c1dcdb4c80aad578","family_id":"qf-density17-counter-restriction-witness","seed":202610177057,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,false,false],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0058","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ea9f2163277ba64aee9d28554b5d083a22a2602fc8a2d51d5416e951a0dd1f7","family_id":"qf-density17-counter-restriction-witness","seed":202610177058,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,false,false],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0059","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac71644c37f49484a6ade20ee95f5258b0f88bbe16f42700eac9041f73c0b929","family_id":"qf-density17-counter-restriction-witness","seed":202610177059,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,false,false],\"witness\":0}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([2, 3, 4], [0, 2, 4], [2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [0, 2, 4], [2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0060","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"857cc55e7e1a225a4cdba6895a37088978f9a2714658e7c3e516efde7e0fb2b4","family_id":"qf-density17-counter-restriction-witness","seed":202610177060,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,true,false,false],\"witness\":1}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0061","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3005266ab791331e321917f6bc08e6645eba18e74fa8af7577d9a9d90f4dbdf4","family_id":"qf-density17-counter-restriction-witness","seed":202610177061,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,false,false],\"witness\":2}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([1, 3, 4], [0, 2, 4], [4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [0, 2, 4], [4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0062","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20818b5500733a7e1e7cd014b0140b8245f187074ccd68377a13d9c339e03efb","family_id":"qf-density17-counter-restriction-witness","seed":202610177062,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,true,false,false],\"witness\":3}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0063","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb3d7ea0be0990fd0b7aa8a2d6e68871744e4ea12abcac80ead1172f36ddef49","family_id":"qf-density17-counter-restriction-witness","seed":202610177063,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,true,false,false],\"witness\":4}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([3, 4], [0, 2, 4], [4], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [0, 2, 4], [4], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0064","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9810bc713d8a3e16b712932cb3f6ebb26d7ea97f570669b8470b304e2787f726","family_id":"qf-density17-counter-restriction-witness","seed":202610177064,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,true,false],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0065","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c9f8dd3f5a4a0808227f8fbeee7eddcb69c99443fbaf0327a07a9414bc5a66","family_id":"qf-density17-counter-restriction-witness","seed":202610177065,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,true,false],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([1, 2, 4], [0, 2, 4], [2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 2, 4], [2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0066","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6262905c731df6fbbb27b57c0b2e9ed8260261787a7c96a76096dc78e9156097","family_id":"qf-density17-counter-restriction-witness","seed":202610177066,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,true,false],\"witness\":0}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0067","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"473944fc2f10060dbb7319bfa176c017672fd56fb8f3dd0a1824b54edc6b8a84","family_id":"qf-density17-counter-restriction-witness","seed":202610177067,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,true,false],\"witness\":1}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([2, 4], [0, 2, 4], [2, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [0, 2, 4], [2, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0068","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3573b07827f10bc921e633b33a915507ab4780883f4a8c6111659fc94a1722c5","family_id":"qf-density17-counter-restriction-witness","seed":202610177068,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,true,true,false],\"witness\":2}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 1, 4], [0, 2, 4], [0, 4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [0, 2, 4], [0, 4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0069","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7d84765a02f2fe4517566e4c382ca79de576a4aadf061340d63835ec4228be","family_id":"qf-density17-counter-restriction-witness","seed":202610177069,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,true,false],\"witness\":3}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 4], [4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 4], [4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0070","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed902f60d51606a6400a969b88ec51a24838c7a6bac3018583e0deba9726dd2f","family_id":"qf-density17-counter-restriction-witness","seed":202610177070,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,true,true,false],\"witness\":4}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0071","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6629b9e747c3f6c7016a00b8487074e6df5847b9ed9323a169cf3d1a670b828c","family_id":"qf-density17-counter-restriction-witness","seed":202610177071,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,true,true,false],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([4], [0, 2, 4], [4], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [0, 2, 4], [4], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0072","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7352439591cfc34c2b003ce5a8d4cc1684e0ce8cf09173748983af8f90567238","family_id":"qf-density17-counter-restriction-witness","seed":202610177072,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,false,true],\"witness\":\"untyped\"}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0073","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"761ffec776e6223b19aca7aff0270de8896bfb8750b0da39cadcf17e15077202","family_id":"qf-density17-counter-restriction-witness","seed":202610177073,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,false,true],\"witness\":0}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([1, 2, 3], [0, 2, 4], [2], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 2, 4], [2], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0074","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff1e4f4c3a083f82141c3b828f98ba4adc46b81f34210f7e9689b71cd5b383e9","family_id":"qf-density17-counter-restriction-witness","seed":202610177074,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,false,true],\"witness\":1}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([0, 2, 3], [0, 2, 4], [0, 2], No, No, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 4], [0, 2], No, No, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0075","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95e2daa7c23dfe884f83056b2b273d4097ce4054b21234b4212bd8a8ab2ecdd8","family_id":"qf-density17-counter-restriction-witness","seed":202610177075,"semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,false,true],\"witness\":2}"},"fields":{"title":"Restriction Can Remove an Original Counterexample","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).","answer":{"kind":"expression","value":"([2, 3], [0, 2, 4], [2], Yes, Yes, No)","explanation":"Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [0, 2, 4], [2], Yes, Yes, No)."},"skills":["Restriction Can Remove an Original Counterexample"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0076","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c00bad0d80e08ea5f3b2a770c214df3704570674d82f6b8516138b16f4632b2","family_id":"qf-density17-counter-domain-extension","seed":202610177076,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":0,\"truth\":[]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([], [[0, Yes]], [], Yes, Yes, 0)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes]], [], Yes, Yes, 0)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0077","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f2215fcb9b1f98c851936e0bbd2fb43946023b2ac2e1491fecd4dafea1ebb3c","family_id":"qf-density17-counter-domain-extension","seed":202610177077,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":0,\"truth\":[]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([], [[0, No]], [0], Yes, No, 0)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, No]], [0], Yes, No, 0)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0078","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1e8f907c611a2f8d351ff7b7fe8e4190b3e078eeb08a6dea4acf516477aec19","family_id":"qf-density17-counter-domain-extension","seed":202610177078,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":8,\"truth\":[true,true,true,true,true,true,true,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[true, true, true, true, true, true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0079","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bc7c0f850639eb6249c2eef1466b6c6ff9d038142290cc20becd14e586672e2","family_id":"qf-density17-counter-domain-extension","seed":202610177079,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":8,\"truth\":[false,false,false,false,false,false,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[false, false, false, false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0080","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a34fab2cb7e3e670842e09817d81caecb8afbd8487291d3f0adea75e666de994","family_id":"qf-density17-counter-domain-extension","seed":202610177080,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0081","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e33c3960858c18e7af4616ffbec6c095c8af4be84e6e325ae1d9f9745ebd0f76","family_id":"qf-density17-counter-domain-extension","seed":202610177081,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0082","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a696f3597d2afafaed781bcf84d56824fa50e69fe99a0df71a74c73ba5ae5f5","family_id":"qf-density17-counter-domain-extension","seed":202610177082,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0083","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb07d29dc3b4fd867e03dfc38894e42e196c24755d70ab44d3e946146633ac53","family_id":"qf-density17-counter-domain-extension","seed":202610177083,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0084","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eb8961dc89b26970dc997873754b794cad45d50b5bf37c8f09b4ee414b89597","family_id":"qf-density17-counter-domain-extension","seed":202610177084,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0085","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0fcad705c8cbe4724a842aa39b2d1f31be6e91690377e40d29fd47bf2cb53db","family_id":"qf-density17-counter-domain-extension","seed":202610177085,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,true,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0086","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c719b222f474a7398d885bcfb1e1ca07fe05a1e1211f1afad0d039eff843df0a","family_id":"qf-density17-counter-domain-extension","seed":202610177086,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,true,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0087","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b31d75c8ce1906ddea93be5b153ddbce45a43ddce2e92a7cb6acd1e546d0240","family_id":"qf-density17-counter-domain-extension","seed":202610177087,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,false,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0088","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f21299dc9d02957b5fe7dc3bb9936444e9ce588e55fde80d904894c211bfcd8f","family_id":"qf-density17-counter-domain-extension","seed":202610177088,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0089","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb54ccfcddf3b2e1306dbaef7c2a66828533c4dc308863375b85f21ca5fbfa3","family_id":"qf-density17-counter-domain-extension","seed":202610177089,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0090","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f51fb683725856ebb8f34853b354f979a0e5ca9f91e410dee5120ee786e6ee3a","family_id":"qf-density17-counter-domain-extension","seed":202610177090,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0091","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3e3c888851a1c4f0d1601c2efd8afa5e7b2e482f6102dcebd075075be0d3f4","family_id":"qf-density17-counter-domain-extension","seed":202610177091,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0092","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed7546fb105d47f03681e80699ccd47512fa84989f7b5ce775cb321a2f28d06","family_id":"qf-density17-counter-domain-extension","seed":202610177092,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0093","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f403166f4325c8b4c8c51fd7466ec1b38b412a9279d3103209e82b73e520ffa1","family_id":"qf-density17-counter-domain-extension","seed":202610177093,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,true,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0094","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"383f3f24406b0a1c9d8c6bb4fd242ba7317410708f7a72e775e8829ad3b80dc2","family_id":"qf-density17-counter-domain-extension","seed":202610177094,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,true,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0095","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61c1c24d16d49fd5d7d47b933df295f77f13b364ca5bf8ea15123cee2aa33f65","family_id":"qf-density17-counter-domain-extension","seed":202610177095,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,true,false]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0096","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e62e3d9e2c191d5571576cc81d094429388748c6b0f731c664aa65c13d842805","family_id":"qf-density17-counter-domain-extension","seed":202610177096,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,false,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0097","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dbbcb45c4d9d5509d7dce6f7dfd29e1a106c74ae092df9e6824b65bffdfac5e","family_id":"qf-density17-counter-domain-extension","seed":202610177097,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,false,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0098","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c07d44ea8dbef9d566a0a0cd3b3853ce0c1d2318a544548a2624a0576bebe6e","family_id":"qf-density17-counter-domain-extension","seed":202610177098,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,false,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0099","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c876173dd1f105052b12d7504edcf2730018ab37c6d37092b5c135ddc6e027","family_id":"qf-density17-counter-domain-extension","seed":202610177099,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,false,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r07-counterexample-0100","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2019a6c91bf06b9c56917e69537f94cf281fdff9121e6e3217d883efdd1cce","family_id":"qf-density17-counter-domain-extension","seed":202610177100,"semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,false,true]}"},"fields":{"title":"One Declared New Predicate Value and Universal Truth","stem":"The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).","answer":{"kind":"expression","value":"([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5)","explanation":"Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5)."},"skills":["One Declared New Predicate Value and Universal Truth"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}