{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r02-counterexample-0001","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c848a39fe8173c81292cf3649fe4fb6b8f3aab51b10ab19a404239846bad0ac9","family_id":"qf-counterexample-witness-validity","seed":202610042001,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[],\"c\":0,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0002","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60a241dac5017e6a7f5efc5be0899359477d7be9394435a016fda697e6e3270e","family_id":"qf-counterexample-witness-validity","seed":202610042002,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-1,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=-1 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0003","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b2e9cc6c049a0e66428a13a9dd404d899d1529d3e43ec7bef67350eaba458b","family_id":"qf-counterexample-witness-validity","seed":202610042003,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":0,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0004","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782e2e71420fc21810e161d1fb637bbd2ed53b11cbc5e4bd3c460aec326fdb53","family_id":"qf-counterexample-witness-validity","seed":202610042004,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-3,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=-3 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0005","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13f832f6c835c056e35e0b817d19b41acb9cd2122724a6463517143904c17fee","family_id":"qf-counterexample-witness-validity","seed":202610042005,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-8,-7,-6],\"c\":-8,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -6} and P(x) means x≥-7. For the claim ∀x∈U P(x), is c=-8 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0006","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fa697a9314b4bc60092db867ab04301197df987b75b08b51adc54f354a5fd7f","family_id":"qf-counterexample-witness-validity","seed":202610042006,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-7,-6,-5],\"c\":-6,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":-6}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -5} and P(x) means x≤-6. For the claim ∀x∈U P(x), is c=-6 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0007","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e575e29c7980be5b017128a33e0f88fadc7e985dffdbcd9918a35db11c29227","family_id":"qf-counterexample-witness-validity","seed":202610042007,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-6,-5,-4],\"c\":-2,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -4} and P(x) means x=-5. For the claim ∀x∈U P(x), is c=-2 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0008","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aced09fdd588aa207d2efb74bb20156051af9b9929f3365b09bd6d19b9a5494c","family_id":"qf-counterexample-witness-validity","seed":202610042008,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-5,-4,-3],\"c\":-5,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -3} and P(x) means x is an integer multiple of (3). For the claim ∀x∈U P(x), is c=-5 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0009","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbae819c864ba5b9cdfb5f8b0ff779e4bbf45fdfa0b909246a68f4ba181912c4","family_id":"qf-counterexample-witness-validity","seed":202610042009,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-4,-3,-2],\"c\":-3,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -2} and P(x) means x≥-3. For the claim ∀x∈U P(x), is c=-3 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0010","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c3b97a0af06cdfa952d1f34d817df69760156af89b44c3e076417f66256f56","family_id":"qf-counterexample-witness-validity","seed":202610042010,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-3,-2,-1],\"c\":1,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":-2}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-3, -2, -1} and P(x) means x≤-2. For the claim ∀x∈U P(x), is c=1 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0011","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ce969e1fd259d3d07f60e1eb56cc6ffac15365f399832e38ce3a145ad154b8","family_id":"qf-counterexample-witness-validity","seed":202610042011,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-2,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x=-1. For the claim ∀x∈U P(x), is c=-2 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0012","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595ec2e0258cb368097f50c4616bc30229d3049ca7fb3739d3230625bd6847cf","family_id":"qf-counterexample-witness-validity","seed":202610042012,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[-1,0,1],\"c\":0,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (4). For the claim ∀x∈U P(x), is c=0 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0013","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5ef3e9eb3e2e90322a8eaa57b3d1ef90e383cac303759fae60ea73cf4b73dad","family_id":"qf-counterexample-witness-validity","seed":202610042013,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[0,1,2],\"c\":4,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={0, 1, 2} and P(x) means x≥1. For the claim ∀x∈U P(x), is c=4 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0014","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f1737a3f94f97bf2334a04d713da05d04235be6de46efe079cb47e31f9caa04","family_id":"qf-counterexample-witness-validity","seed":202610042014,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[1,2,3],\"c\":1,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":2}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≤2. For the claim ∀x∈U P(x), is c=1 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0015","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4058f401d063d4ec1fa16c69c5a883d3b0d2d211992578488727bb86f9b5b2b","family_id":"qf-counterexample-witness-validity","seed":202610042015,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[2,3,4],\"c\":3,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={2, 3, 4} and P(x) means x=3. For the claim ∀x∈U P(x), is c=3 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0016","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eadfa14a8302c5fb02da15acda2814f04b104fecfd73c274d18b23a7768ca480","family_id":"qf-counterexample-witness-validity","seed":202610042016,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[3,4,5],\"c\":7,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x is an integer multiple of (2). For the claim ∀x∈U P(x), is c=7 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0017","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"988ba87afdd902b61e168acd042d9dd04a39aa051d5cd9dd9b8616e6394defc0","family_id":"qf-counterexample-witness-validity","seed":202610042017,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[4,5,6],\"c\":4,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={4, 5, 6} and P(x) means x≥5. For the claim ∀x∈U P(x), is c=4 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0018","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e93c6debbf755f495c78d2c32e19c3bc7e575b0fa102b3cf852ad36f1d3fa45","family_id":"qf-counterexample-witness-validity","seed":202610042018,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[5,6,7],\"c\":6,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":6}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≤6. For the claim ∀x∈U P(x), is c=6 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0019","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1355c689174ae9c43eae80fb1e0627b40798480155be0b13dc5901fcefe450d1","family_id":"qf-counterexample-witness-validity","seed":202610042019,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[6,7,8],\"c\":10,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={6, 7, 8} and P(x) means x=7. For the claim ∀x∈U P(x), is c=10 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0020","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56cfc6475ba48d204380513d7c370541ea7ff6e36435759a99ac6247f5c94ef","family_id":"qf-counterexample-witness-validity","seed":202610042020,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[7,8,9],\"c\":7,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={7, 8, 9} and P(x) means x is an integer multiple of (3). For the claim ∀x∈U P(x), is c=7 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0021","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab9b91689fe3102e093f7e95fd6c0b5d71948fbc0cc2838a8b248530c315fac9","family_id":"qf-counterexample-witness-validity","seed":202610042021,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[8,9,10],\"c\":9,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={8, 9, 10} and P(x) means x≥9. For the claim ∀x∈U P(x), is c=9 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0022","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7654dd400862b131679823283b7d1eb39cc0df2dddc39e04b35c3e4277985cfa","family_id":"qf-counterexample-witness-validity","seed":202610042022,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[9,10,11],\"c\":13,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":10}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={9, 10, 11} and P(x) means x≤10. For the claim ∀x∈U P(x), is c=13 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0023","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2640cf180b820a7c23a282415d936e5aacc0d265972e52a9391c36ec3a9f570","family_id":"qf-counterexample-witness-validity","seed":202610042023,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[10,11,12],\"c\":10,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x=11. For the claim ∀x∈U P(x), is c=10 a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0024","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4a09cfa65d1697fb4c416ad58f1899784e3534d1c8737e7c9709ba6b115b650","family_id":"qf-counterexample-witness-validity","seed":202610042024,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[11,12,13],\"c\":12,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={11, 12, 13} and P(x) means x is an integer multiple of (4). For the claim ∀x∈U P(x), is c=12 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0025","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ca63ff5a202d9a6db1848d9dd8dc83b4a7ff7df6ed9c5f36602a8e7938f16be","family_id":"qf-counterexample-witness-validity","seed":202610042025,"semantic_template":"qf-counterexample-witness-validity/{\"U\":[12,13,14],\"c\":16,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Domain Eligibility and Predicate Failure","stem":"Using classical truth semantics, the explicit finite integer domain is U={12, 13, 14} and P(x) means x≥13. For the claim ∀x∈U P(x), is c=16 a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Domain Eligibility and Predicate Failure"],"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-20261003-advance-20261004-r02-counterexample-0026","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5fecf96a06c3eeaf3252a3ec40f088f9c138c0444ac616b3bf79a2b782e9ace","family_id":"qf-counterexample-first-witness","seed":202610042026,"semantic_template":"qf-counterexample-first-witness/{\"U\":[],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"none","explanation":"The complete counterexample set is ∅. It is empty, so there is no counterexample in U. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0027","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"608077792e0cbd8c558019d9a74171c62fd8eb9a94e6c938cfa55baa58da7a0c","family_id":"qf-counterexample-first-witness","seed":202610042027,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-2,-1,0],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"none","explanation":"The complete counterexample set is ∅. It is empty, so there is no counterexample in U. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0028","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b077030b174adad0fd31767f61182e1a411ee04aa9ee815704708efc09986d45","family_id":"qf-counterexample-first-witness","seed":202610042028,"semantic_template":"qf-counterexample-first-witness/{\"U\":[1,2,3],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"1","explanation":"The complete counterexample set is {1, 2, 3}. Its least element is 1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0029","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d74ae7d8136f8f40bf4ad2364b6a81992277f590825b6c9bafda599343524e8","family_id":"qf-counterexample-first-witness","seed":202610042029,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-1,0,1],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-1","explanation":"The complete counterexample set is {-1, 1}. Its least element is -1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0030","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30834cc01cc343522d0207ed6a6f3da0ed1ea3bf51768bb9a1ca21599eced084","family_id":"qf-counterexample-first-witness","seed":202610042030,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-8,-7,-5,-3],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-8","explanation":"The complete counterexample set is {-8}. Its least element is -8. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0031","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a8da269e3b8390be41fcaf4250eaea6fa898345e1820b8963a87240ee74e293","family_id":"qf-counterexample-first-witness","seed":202610042031,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-7,-6,-4,-2],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-2","explanation":"The complete counterexample set is {-2}. Its least element is -2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0032","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0f19da8c5fe0ba6b9fcdc0a35b7730412662db03d839ee7d61d513d8c78813","family_id":"qf-counterexample-first-witness","seed":202610042032,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-6,-5,-3,-1],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-6","explanation":"The complete counterexample set is {-6, -3, -1}. Its least element is -6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0033","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74a220ddcdc019132c8badea4000bc726037848ffc20a9b7e60d3d51d7945a2","family_id":"qf-counterexample-first-witness","seed":202610042033,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-5,-4,-2,0],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -2, 0} and P(x) means x is an integer multiple of (3). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-5","explanation":"The complete counterexample set is {-5, -4, -2}. Its least element is -5. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0034","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ae61624ba9f350da6179d579e8bb645a403d9ed0d6555b90d83e293da93e672","family_id":"qf-counterexample-first-witness","seed":202610042034,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-4,-3,-1,1],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-4","explanation":"The complete counterexample set is {-4}. Its least element is -4. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0035","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc44d95d33aeebfd12ff45548855d814820387808730c91446e4dc5657b98ee3","family_id":"qf-counterexample-first-witness","seed":202610042035,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-3,-2,0,2],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"2","explanation":"The complete counterexample set is {2}. Its least element is 2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0036","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b41c00258bd744e3f162f161e6941e437ac3dfedaac8dc9ff136f19c6a413e7","family_id":"qf-counterexample-first-witness","seed":202610042036,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-2,-1,1,3],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-2","explanation":"The complete counterexample set is {-2, 1, 3}. Its least element is -2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0037","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5496b9acb3118aa3b6f11b24401db75533ea5e861c368abb214d763542b9597e","family_id":"qf-counterexample-first-witness","seed":202610042037,"semantic_template":"qf-counterexample-first-witness/{\"U\":[-1,0,2,4],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 2, 4} and P(x) means x is an integer multiple of (4). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"-1","explanation":"The complete counterexample set is {-1, 2}. Its least element is -1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0038","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"812250036d0dd7496925df118c4636a4b97d030991611be8a9618f2c5055f5be","family_id":"qf-counterexample-first-witness","seed":202610042038,"semantic_template":"qf-counterexample-first-witness/{\"U\":[0,1,3,5],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"0","explanation":"The complete counterexample set is {0}. Its least element is 0. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0039","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bcdee364dc551907e8f31e4b4bb227bbe33e9147d0c16553ee53aed2bc64420","family_id":"qf-counterexample-first-witness","seed":202610042039,"semantic_template":"qf-counterexample-first-witness/{\"U\":[1,2,4,6],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"6","explanation":"The complete counterexample set is {6}. Its least element is 6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0040","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36ca3baedf362e97e6ba5a28224701ff2493826609d4efedd1b1393f6826c0dc","family_id":"qf-counterexample-first-witness","seed":202610042040,"semantic_template":"qf-counterexample-first-witness/{\"U\":[2,3,5,7],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"2","explanation":"The complete counterexample set is {2, 5, 7}. Its least element is 2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0041","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cd9c0a27e625329bb41957e4d1075bdc54248d38d816fab38e55787b3dfa22a","family_id":"qf-counterexample-first-witness","seed":202610042041,"semantic_template":"qf-counterexample-first-witness/{\"U\":[3,4,6,8],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={3, 4, 6, 8} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"3","explanation":"The complete counterexample set is {3}. Its least element is 3. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0042","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e537af4934aeb60f295ca5a9caff09cee6cd17ebfca51f427f2d121a0b5d64cc","family_id":"qf-counterexample-first-witness","seed":202610042042,"semantic_template":"qf-counterexample-first-witness/{\"U\":[4,5,7,9],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"4","explanation":"The complete counterexample set is {4}. Its least element is 4. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0043","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"407245ac1e3a72ef78f346efc0a2fe4e5ccc43eea9ecb161d7cd00ff992801de","family_id":"qf-counterexample-first-witness","seed":202610042043,"semantic_template":"qf-counterexample-first-witness/{\"U\":[5,6,8,10],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"10","explanation":"The complete counterexample set is {10}. Its least element is 10. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0044","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9264a7ded2c6d56600242e5ccc8f1127fff840be95ddfeef7aa356552cfd1e91","family_id":"qf-counterexample-first-witness","seed":202610042044,"semantic_template":"qf-counterexample-first-witness/{\"U\":[6,7,9,11],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"6","explanation":"The complete counterexample set is {6, 9, 11}. Its least element is 6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0045","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b170da87399a160d0b7e488983be6c3e12bace5627faea46a4aa7d7681eb918","family_id":"qf-counterexample-first-witness","seed":202610042045,"semantic_template":"qf-counterexample-first-witness/{\"U\":[7,8,10,12],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={7, 8, 10, 12} and P(x) means x is an integer multiple of (3). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"7","explanation":"The complete counterexample set is {7, 8, 10}. Its least element is 7. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0046","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95367f9237d974df2755eaaedb512713d9e123cfdb5f990590e82406e38a5dda","family_id":"qf-counterexample-first-witness","seed":202610042046,"semantic_template":"qf-counterexample-first-witness/{\"U\":[8,9,11,13],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"8","explanation":"The complete counterexample set is {8}. Its least element is 8. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0047","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b9ffad09d1eeb454053d5f18630176fcd5eca0ca58db6cf3533ebf7c050f121","family_id":"qf-counterexample-first-witness","seed":202610042047,"semantic_template":"qf-counterexample-first-witness/{\"U\":[9,10,12,14],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"14","explanation":"The complete counterexample set is {14}. Its least element is 14. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0048","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"943457af52678b6d4908f9cb04a7a02019ec1f325918b1dda57f6be971497419","family_id":"qf-counterexample-first-witness","seed":202610042048,"semantic_template":"qf-counterexample-first-witness/{\"U\":[10,11,13,15],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"10","explanation":"The complete counterexample set is {10, 13, 15}. Its least element is 10. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0049","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8727da34f545a1499f8eb260b27b5db3ffbc32ef0c8f0dbc849e0a36e83ac819","family_id":"qf-counterexample-first-witness","seed":202610042049,"semantic_template":"qf-counterexample-first-witness/{\"U\":[11,12,14,16],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14, 16} and P(x) means x is an integer multiple of (4). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"11","explanation":"The complete counterexample set is {11, 14}. Its least element is 11. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0050","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b395599b000ec6a7481588c081b2fa260f888d65f93001a84dfaf712e264432","family_id":"qf-counterexample-first-witness","seed":202610042050,"semantic_template":"qf-counterexample-first-witness/{\"U\":[12,13,15,17],\"operation\":\"first_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Least Counterexample in a Finite Domain","stem":"Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.","answer":{"kind":"short_text","value":"12","explanation":"The complete counterexample set is {12}. Its least element is 12. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Least Counterexample in a Finite Domain"],"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-20261003-advance-20261004-r02-counterexample-0051","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c60bf131f9dbddf616daa21da8f2d98cd16b81257a71e98ab801e735b0a96ad8","family_id":"qf-counterexample-candidate-set","seed":202610042051,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-2,0,4],\"U\":[],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":0}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x=0. Which members of C={-2, 0, 4} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is ∅; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0052","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79803bd0e6b1a4c9bc4521ab2359ff9b50a6f7ecd4bd6a51eb92b3c90941f184","family_id":"qf-counterexample-candidate-set","seed":202610042052,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-3,-2,0,5],\"U\":[-2,-1,0],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Which members of C={-3, -2, 0, 5} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is ∅; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0053","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"058418349d5b43d79d5a6a1064162c664aa35c830b399ab686577ddca1f32b8a","family_id":"qf-counterexample-candidate-set","seed":202610042053,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-12,-10,-9,-7,-3],\"U\":[-10,-9,-7],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":-9}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -7} and P(x) means x=-9. Which members of C={-12, -10, -9, -7, -3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-10, -7}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-10, -7}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0054","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4192a1496b6a8c460bb84e268f9e5bcceff90aedcff214f6ce4e6028cd9c3378","family_id":"qf-counterexample-candidate-set","seed":202610042054,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-11,-9,-8,-6,-2],\"U\":[-9,-8,-6],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-9, -8, -6} and P(x) means x is an integer multiple of (2). Which members of C={-11, -9, -8, -6, -2} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-9}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-9}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0055","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7c551922e7a3b4071f8b6c5a0d1c6a2f2dfb7cacfba1c5e0b9f18ad73b5e8eb","family_id":"qf-counterexample-candidate-set","seed":202610042055,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-10,-8,-7,-5,-1],\"U\":[-8,-7,-5],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5} and P(x) means x≥-7. Which members of C={-10, -8, -7, -5, -1} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-8}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0056","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7368244663ccc4cc8e9a740699dabf67939f1b7cbe138412280911d97fb73d12","family_id":"qf-counterexample-candidate-set","seed":202610042056,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-9,-7,-6,-4,0],\"U\":[-7,-6,-4],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":-6}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4} and P(x) means x≤-6. Which members of C={-9, -7, -6, -4, 0} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-4}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0057","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"110207e8f0a7704915661dd2983d972ca55555d09d1cb663019e86cb7d6b6ed1","family_id":"qf-counterexample-candidate-set","seed":202610042057,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-8,-6,-5,-3,1],\"U\":[-6,-5,-3],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3} and P(x) means x=-5. Which members of C={-8, -6, -5, -3, 1} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-6, -3}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-6, -3}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0058","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c6e46f1dad1e825206a90ff42ff9ebb00505897e3ab46066e625ce4009e32c7","family_id":"qf-counterexample-candidate-set","seed":202610042058,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-7,-5,-4,-2,2],\"U\":[-5,-4,-2],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -2} and P(x) means x is an integer multiple of (3). Which members of C={-7, -5, -4, -2, 2} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-5, -4, -2}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-5, -4, -2}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0059","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28287c9cbe635b8c2d94e7edbcea7ebba4b0ef51b5061233e8ea3c8e897ad386","family_id":"qf-counterexample-candidate-set","seed":202610042059,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-6,-4,-3,-1,3],\"U\":[-4,-3,-1],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1} and P(x) means x≥-3. Which members of C={-6, -4, -3, -1, 3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-4}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0060","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4161cf76a37941c0f7ced57acd2f303db88c9b55db64bac84f23f2ecc1b330f","family_id":"qf-counterexample-candidate-set","seed":202610042060,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-5,-3,-2,0,4],\"U\":[-3,-2,0],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":-2}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0} and P(x) means x≤-2. Which members of C={-5, -3, -2, 0, 4} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{0}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {0}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0061","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fa9fcc43692382e65d081d6c854044f1d923ce7c26dae83a5004c39e523facd","family_id":"qf-counterexample-candidate-set","seed":202610042061,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-4,-2,-1,1,5],\"U\":[-2,-1,1],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1} and P(x) means x=-1. Which members of C={-4, -2, -1, 1, 5} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-2, 1}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-2, 1}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0062","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8f38c390f20e92295b44342d4b735ea25ae7a8d282d3f80ee22690a9eb3d4cb","family_id":"qf-counterexample-candidate-set","seed":202610042062,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-3,-1,0,2,6],\"U\":[-1,0,2],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 2} and P(x) means x is an integer multiple of (4). Which members of C={-3, -1, 0, 2, 6} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-1, 2}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {-1, 2}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0063","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8288020149e8c2f34100e6c31f8cc8ea8bc50b210f961aeabba24129f94f6361","family_id":"qf-counterexample-candidate-set","seed":202610042063,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-2,0,1,3,7],\"U\":[0,1,3],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3} and P(x) means x≥1. Which members of C={-2, 0, 1, 3, 7} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{0}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {0}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0064","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5f88936f7768fc9a7112a93ed7c6bc5610ff00c2cd29673e279c9f953d9c5dc","family_id":"qf-counterexample-candidate-set","seed":202610042064,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[-1,1,2,4,8],\"U\":[1,2,4],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":2}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4} and P(x) means x≤2. Which members of C={-1, 1, 2, 4, 8} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{4}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0065","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"003764cde15b3865e493802fe1bfb518bfab66eebcc32d42b1b9bfdcb89c6d81","family_id":"qf-counterexample-candidate-set","seed":202610042065,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[0,2,3,5,9],\"U\":[2,3,5],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5} and P(x) means x=3. Which members of C={0, 2, 3, 5, 9} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{2, 5}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {2, 5}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0066","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e466ab4730d40792fd7913614ca3cc26c4083d7db7a9dcc41ad779a64a14ef","family_id":"qf-counterexample-candidate-set","seed":202610042066,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[1,3,4,6,10],\"U\":[3,4,6],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={3, 4, 6} and P(x) means x is an integer multiple of (2). Which members of C={1, 3, 4, 6, 10} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{3}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {3}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0067","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ef1fbab814f0dee25f63c07dec03fd65620dd2b3405204e888dc9e17624826","family_id":"qf-counterexample-candidate-set","seed":202610042067,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[2,4,5,7,11],\"U\":[4,5,7],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7} and P(x) means x≥5. Which members of C={2, 4, 5, 7, 11} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{4}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0068","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28b343212072c512bd7cf954c1cde967e1cd39397cdd6c60c8c200393fe637fe","family_id":"qf-counterexample-candidate-set","seed":202610042068,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[3,5,6,8,12],\"U\":[5,6,8],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":6}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8} and P(x) means x≤6. Which members of C={3, 5, 6, 8, 12} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{8}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0069","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"497f50fa92bbe72a2914e8e9ff2fdf11787896fd13944fe215038f054923f4d2","family_id":"qf-counterexample-candidate-set","seed":202610042069,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[4,6,7,9,13],\"U\":[6,7,9],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9} and P(x) means x=7. Which members of C={4, 6, 7, 9, 13} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{6, 9}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {6, 9}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0070","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59968a62b6e34a8fdce826bb49b317449b106b177ca56c1eb5b1a0b39dd22a6c","family_id":"qf-counterexample-candidate-set","seed":202610042070,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[5,7,8,10,14],\"U\":[7,8,10],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={7, 8, 10} and P(x) means x is an integer multiple of (3). Which members of C={5, 7, 8, 10, 14} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{7, 8, 10}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {7, 8, 10}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0071","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fa3fefc6e564a0d4d3b4dfcfea17ce4e58a2417aaaea120a85325dd00754973","family_id":"qf-counterexample-candidate-set","seed":202610042071,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[6,8,9,11,15],\"U\":[8,9,11],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{8}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0072","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87842e37c38389749965e8b6f48a851cf5afcb5cd1fefd3da0e252216bd93d38","family_id":"qf-counterexample-candidate-set","seed":202610042072,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[7,9,10,12,16],\"U\":[9,10,12],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":10}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12} and P(x) means x≤10. Which members of C={7, 9, 10, 12, 16} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{12}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {12}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0073","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9840cf02e2c9b5cf4535f65fc07405f6580f638cdaf939192417e38f104e4412","family_id":"qf-counterexample-candidate-set","seed":202610042073,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[8,10,11,13,17],\"U\":[10,11,13],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13} and P(x) means x=11. Which members of C={8, 10, 11, 13, 17} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{10, 13}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {10, 13}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0074","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"876e71917134ff49184db606efc2abe862000398435c9df8ba2af015a87ea8bf","family_id":"qf-counterexample-candidate-set","seed":202610042074,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[9,11,12,14,18],\"U\":[11,12,14],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14} and P(x) means x is an integer multiple of (4). Which members of C={9, 11, 12, 14, 18} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{11, 14}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {11, 14}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0075","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2bb7c36339693ae7a6c82e0da32cf443664f04c8264fad9fcd0c7b5f05bd117","family_id":"qf-counterexample-candidate-set","seed":202610042075,"semantic_template":"qf-counterexample-candidate-set/{\"C\":[10,12,13,15,19],\"U\":[12,13,15],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Valid Counterexamples in a Candidate Set","stem":"Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15} and P(x) means x≥13. Which members of C={10, 12, 13, 15, 19} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.","answer":{"kind":"expression","value":"{12}","explanation":"Require membership in both U and C and failure of P. The valid candidate counterexample set is {12}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Valid Counterexamples in a Candidate Set"],"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-20261003-advance-20261004-r02-counterexample-0076","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475038a70833a9fb6f2cdd97973b493131258841ec37e0f9d01577fdb11c9b78","family_id":"qf-counterexample-domain-restriction","seed":202610042076,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-12,-11,-10],\"V\":[-12],\"c\":-12,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-11}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-12, -11, -10} and P(x) means x≥-11. The value c=-12 is a valid counterexample to the original universal claim. After restricting to V={-12}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0077","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d503d46306569cd1f994eaf99bc61b17b36881127c57f21305155ef3b069f5c8","family_id":"qf-counterexample-domain-restriction","seed":202610042077,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-11,-10,-9],\"V\":[-10,-9],\"c\":-11,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-9}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-11, -10, -9} and P(x) means x≥-9. The value c=-11 is a valid counterexample to the original universal claim. After restricting to V={-10, -9}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0078","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e26193d0b0d7aa8a1aabeface60284fdaf583dd97a119b97ce7766fa56e5508","family_id":"qf-counterexample-domain-restriction","seed":202610042078,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-10,-9,-8],\"V\":[],\"c\":-10,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-9}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -8} and P(x) means x≥-9. The value c=-10 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0079","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df7d6dca6cfdc33bd891bf4eb777cac63f57fb77e6f5d83d22fd9e69e6d883cd","family_id":"qf-counterexample-domain-restriction","seed":202610042079,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-9,-8,-7],\"V\":[-9],\"c\":-9,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-9, -8, -7} and P(x) means x≥-7. The value c=-9 is a valid counterexample to the original universal claim. After restricting to V={-9}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0080","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08a6bb8385015821b1f22be92b69ec2af65663327441ad347850d0cacd61b479","family_id":"qf-counterexample-domain-restriction","seed":202610042080,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-8,-7,-6],\"V\":[-7,-6],\"c\":-8,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -6} and P(x) means x≥-7. The value c=-8 is a valid counterexample to the original universal claim. After restricting to V={-7, -6}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0081","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f0134289f592cfc5f048cd01ee26884438f33b93299997b1e5b4884642d61b8","family_id":"qf-counterexample-domain-restriction","seed":202610042081,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-7,-6,-5],\"V\":[],\"c\":-7,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-5}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -5} and P(x) means x≥-5. The value c=-7 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0082","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b0ac2a6a0eb76f4dc2d7a1b233c3f2763de6ad16830a515835b4cc3ff7b4d8","family_id":"qf-counterexample-domain-restriction","seed":202610042082,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-6,-5,-4],\"V\":[-6],\"c\":-6,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-5}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -4} and P(x) means x≥-5. The value c=-6 is a valid counterexample to the original universal claim. After restricting to V={-6}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0083","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a3e8de71ad11c9a9503193b54183a4779e59b15309052e2b636e3c927dbbe9","family_id":"qf-counterexample-domain-restriction","seed":202610042083,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-5,-4,-3],\"V\":[-4,-3],\"c\":-5,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -3} and P(x) means x≥-3. The value c=-5 is a valid counterexample to the original universal claim. After restricting to V={-4, -3}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0084","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96237ff7fef794b53b829db8c37e0dde502f4cc21febee3f18dccd77927a590f","family_id":"qf-counterexample-domain-restriction","seed":202610042084,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-4,-3,-2],\"V\":[],\"c\":-4,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -2} and P(x) means x≥-3. The value c=-4 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0085","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"214fb37f8367ddea05927a17f6177413ade8ace236a2796ad01d39ad170db0b1","family_id":"qf-counterexample-domain-restriction","seed":202610042085,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-3,-2,-1],\"V\":[-3],\"c\":-3,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-1}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-3, -2, -1} and P(x) means x≥-1. The value c=-3 is a valid counterexample to the original universal claim. After restricting to V={-3}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0086","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f4b3e5a4a677bfc75e70619f762839b671c0b7cfccd7142e8946797490fc6a","family_id":"qf-counterexample-domain-restriction","seed":202610042086,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-2,-1,0],\"V\":[-1,0],\"c\":-2,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":-1}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-1. The value c=-2 is a valid counterexample to the original universal claim. After restricting to V={-1, 0}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0087","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54a38bdb46cc9dc711c5fe30968776ffc329dbbb57e045f375c6cb4c4fe869ea","family_id":"qf-counterexample-domain-restriction","seed":202610042087,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-1,0,1],\"V\":[],\"c\":-1,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x≥1. The value c=-1 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0088","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91210cc239cf76ee6d57b6799722becab8bea10a4df9dd6ddc5e81c61a4dd46f","family_id":"qf-counterexample-domain-restriction","seed":202610042088,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[0,1,2],\"V\":[0],\"c\":0,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={0, 1, 2} and P(x) means x≥1. The value c=0 is a valid counterexample to the original universal claim. After restricting to V={0}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0089","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d52ea2200c24f623abfb331c631edb2fe718d9d78e4225632a00d1d5a7dade","family_id":"qf-counterexample-domain-restriction","seed":202610042089,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[1,2,3],\"V\":[2,3],\"c\":1,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":3}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥3. The value c=1 is a valid counterexample to the original universal claim. After restricting to V={2, 3}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0090","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ece3aa1c4162a2de9b576631ca20f82cf8617da37ac29d4dda42f0239cecc5e7","family_id":"qf-counterexample-domain-restriction","seed":202610042090,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[2,3,4],\"V\":[],\"c\":2,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":3}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={2, 3, 4} and P(x) means x≥3. The value c=2 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0091","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad8b5b1c3d547438d32d3e129100f17640d8550679fb4a200f361aad82c1f1a9","family_id":"qf-counterexample-domain-restriction","seed":202610042091,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[3,4,5],\"V\":[3],\"c\":3,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x≥5. The value c=3 is a valid counterexample to the original universal claim. After restricting to V={3}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0092","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d246a0ee3bd41069c15eaf312fefad6ca5d438fa223c43069978c3a0d03633a3","family_id":"qf-counterexample-domain-restriction","seed":202610042092,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[4,5,6],\"V\":[5,6],\"c\":4,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={4, 5, 6} and P(x) means x≥5. The value c=4 is a valid counterexample to the original universal claim. After restricting to V={5, 6}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0093","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a269dc0829bc750ebe4b5bece4d78a7ec2fde7b63f9b920097da5ccc027114c3","family_id":"qf-counterexample-domain-restriction","seed":202610042093,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[5,6,7],\"V\":[],\"c\":5,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":7}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≥7. The value c=5 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0094","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"074f637d3419bd30508263d8eb31d562bb11662e72a80be082b4a0cfa5ea1893","family_id":"qf-counterexample-domain-restriction","seed":202610042094,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[6,7,8],\"V\":[6],\"c\":6,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":7}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={6, 7, 8} and P(x) means x≥7. The value c=6 is a valid counterexample to the original universal claim. After restricting to V={6}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0095","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4db83cbf2217c4e3c075a188ff756a59fb622f9fedad3f487cee0fdeaf850f67","family_id":"qf-counterexample-domain-restriction","seed":202610042095,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[7,8,9],\"V\":[8,9],\"c\":7,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={7, 8, 9} and P(x) means x≥9. The value c=7 is a valid counterexample to the original universal claim. After restricting to V={8, 9}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0096","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ea383c049979d4dce12d5561dbeb00ecc593ae14cc51bd1ffc88041f491ec6","family_id":"qf-counterexample-domain-restriction","seed":202610042096,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[8,9,10],\"V\":[],\"c\":8,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={8, 9, 10} and P(x) means x≥9. The value c=8 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0097","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0c1ebbce46f96fec97ebfd038582c723c58f807e083c03511b1826574aab6fc","family_id":"qf-counterexample-domain-restriction","seed":202610042097,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[9,10,11],\"V\":[9],\"c\":9,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":11}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={9, 10, 11} and P(x) means x≥11. The value c=9 is a valid counterexample to the original universal claim. After restricting to V={9}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0098","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1255dfd7ad23701426b1819087e43e7ce45267612a17d643c685139adb46ea0d","family_id":"qf-counterexample-domain-restriction","seed":202610042098,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[10,11,12],\"V\":[11,12],\"c\":10,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":11}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x≥11. The value c=10 is a valid counterexample to the original universal claim. After restricting to V={11, 12}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0099","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"386db849e9a78b88034164011a2c2d6a54917e4113b9b66ea79143263cfe9e7f","family_id":"qf-counterexample-domain-restriction","seed":202610042099,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[11,12,13],\"V\":[],\"c\":11,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={11, 12, 13} and P(x) means x≥13. The value c=11 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"No","explanation":"The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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-20261003-advance-20261004-r02-counterexample-0100","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9a101ddbef38563ab7c5b63c16a4f50717c23bcce3477beebab76c06438684a","family_id":"qf-counterexample-domain-restriction","seed":202610042100,"semantic_template":"qf-counterexample-domain-restriction/{\"U\":[12,13,14],\"V\":[12],\"c\":12,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Counterexample Eligibility after Domain Restriction","stem":"Using classical truth semantics, the explicit finite integer domain is U={12, 13, 14} and P(x) means x≥13. The value c=12 is a valid counterexample to the original universal claim. After restricting to V={12}⊆U, is c still a valid counterexample?","answer":{"kind":"short_text","value":"Yes","explanation":"The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."},"skills":["Counterexample Eligibility after Domain Restriction"],"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."}}