{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r03-proposition-0001","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b91211b327331b2f1e7aa34b99fd2037378a6f0fd70b4d133ef7326b4a40669","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123001,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"0\",\"-2\",\"2\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [0, -2, 2])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [0, -2, 2])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0002","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc4e089d1f4819f5eec18bb3be61147eae86355e856b3d2593c32db94fc2d0bb","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123002,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-11\",\"-2\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-11, -2, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-11, -2, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0003","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f17b146aee2a5c34d2da92df2b5c940c2f662d40e3eb3ae8c111ca919e6079c","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123003,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-10\",\"-3\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [-10, -3, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-10, -3, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0004","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98d26a9f30dfe5db6360d1cc626566104d24c759aa7189125bc2843ec8cec23a","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123004,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-9\",\"-4\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-9, -4, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-9, -4, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0005","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbba871deda53759b78af6b5b4d2b7064edb211764f54243e8943bf05539238d","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123005,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-8\",\"-5\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-8, -5, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-8, -5, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0006","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a67879b58448c37f905c6c7e2d932affd512650bebf9541bce900142ecad7cbe","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123006,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-7\",\"-6\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [-7, -6, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-7, -6, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0007","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f056ada29b21b7250b9a90542a233a62c692720da08b2ac749875ca487a0b13","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123007,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-6\",\"-7\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-6, -7, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-6, -7, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0008","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b6755bec757285185480aae629da331ff038949b93de83a19de5f34493216b5","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123008,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-5\",\"-8\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-5, -8, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-5, -8, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0009","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9492524bc0b28838959f69bafcd5a3703d7200cb239978bee7e02c718a3b261d","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123009,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-4\",\"-9\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [-4, -9, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-4, -9, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0010","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e6c587b807ec33ad7070909c75b2189e46b7ad1bc39a1bae358e59b62bc3e15","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123010,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-3\",\"-10\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-3, -10, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-3, -10, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0011","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ef241b8d1a1c8b593ac38e797b7976a7e4e478e2c265a876ba3915def906ac","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123011,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-2\",\"-11\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [-2, -11, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-2, -11, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0012","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb042a029abb75c04a5714b53f827530523f19a514b84b514813b678ff4f6d64","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123012,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-1\",\"-12\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [-1, -12, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-1, -12, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0013","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31eaec970163eb781e7e11dbcf33e51fe396d99a34810bc80a0ac45ffd32b11d","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123013,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"0\",\"-13\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [0, -13, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [0, -13, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0014","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"385e28b8287a3cdc483fff336cf93b4a865fd6e272d4c7b661cc118064cc9fc7","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123014,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"1\",\"-14\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [1, -14, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [1, -14, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0015","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77adae05a6ef0a71fe99463fd5b06783468a9b64c4a67f418557959b0f2df9b8","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123015,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"2\",\"-15\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [2, -15, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [2, -15, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0016","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44b2b41834dc828e5e4cb0ad42247ab6f3f7827f2d6b876427552689dd87c22a","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123016,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"3\",\"-16\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [3, -16, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [3, -16, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0017","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac11d6eaffc46f07f77ba9558ea0b8bbb24a1b29f405dc91593cdca4af1cf404","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123017,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"4\",\"-17\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [4, -17, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [4, -17, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0018","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb7fa0b942f7720a601ed2159b0ddac1c33668996449402eb666150f3f61d66","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123018,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"5\",\"-18\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [5, -18, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [5, -18, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0019","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec36c3c1f1942439adfd0cc02974c8a9fb17d364c844c4e8eda2ced7b711bc45","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123019,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"6\",\"-19\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [6, -19, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [6, -19, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0020","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc168b01d0787990646b07360d02f41f0e6d9b2006938a8f7d7bc89dd31d595b","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123020,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"7\",\"-20\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [7, -20, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [7, -20, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0021","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1402bdd13dc08fe2c1ad890bdfa5800d81238365ae63c4cd76987bd267bee8ec","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123021,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"8\",\"-21\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [8, -21, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [8, -21, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0022","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f15c1be8f260e896fbc7ae84a37c7a16bee884d21f47047440966658389c7e7c","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123022,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"9\",\"-22\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [9, -22, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [9, -22, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0023","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"007dabfb280c708cba592f6307aee068365e0cf9906021f883ff6de41bda98ed","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123023,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"10\",\"-23\",\"0\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [10, -23, 0])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [10, -23, 0])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0024","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e14a3918488725547ad1fbbe6514601a5683ca0de8fa4ed258636123056a0902","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123024,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"11\",\"-24\",\"1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [11, -24, 1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [11, -24, 1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0025","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b947ba9d268ecb8c3aae8fdaaf6eda167ee377847e332211b61e63f175cddb8","family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123025,"semantic_template":"qf-density12-proposition-comparison-witnesses/[\"12\",\"-25\",\"-1\"]"},"fields":{"title":"Closed Comparison Truths and Signed Witnesses","stem":"In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [12, -25, -1])","explanation":"Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [12, -25, -1])."},"skills":["Closed Comparison Truths and Signed Witnesses"],"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-20261008-advance-20261012-r03-proposition-0026","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0213658fad3206e7b0ee2063567e34f2c15c4d2cf79a36b1187d7704aa0d11e5","family_id":"qf-density12-proposition-open-binding","seed":202610123026,"semantic_template":"qf-density12-proposition-open-binding/[\"open_sentence\"]"},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-2 with assignment=null (unbound). Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(No, open, not-applicable)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (No, open, not-applicable)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0027","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d7d392d7d2d0c4d7057478c864728aea67164c7c158137fe1da8c0c91c31fad","family_id":"qf-density12-proposition-open-binding","seed":202610123027,"semantic_template":"qf-density12-proposition-open-binding/\"-11\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-1 with assignment=-12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -11)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -11)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0028","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c271b9b3162d7892c309c53bd1826ec3defe8204899b97a2b9abce98c5abbcb2","family_id":"qf-density12-proposition-open-binding","seed":202610123028,"semantic_template":"qf-density12-proposition-open-binding/\"-10\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤0 with assignment=-10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -10)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -10)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0029","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"133071ea884454e1ecee6fbbb49c402ad54c1f9a0e3ea37b12be98a54dad7978","family_id":"qf-density12-proposition-open-binding","seed":202610123029,"semantic_template":"qf-density12-proposition-open-binding/\"-9\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤1 with assignment=-8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -9)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -9)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0030","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e5294001963dbf70ec4c26cee69f38e889be94cbf9ea9fff03a4a0110c10eab","family_id":"qf-density12-proposition-open-binding","seed":202610123030,"semantic_template":"qf-density12-proposition-open-binding/\"-8\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤2 with assignment=-6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -8)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -8)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0031","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486fd2318cf3af75c246c1ddc8ae97e01069c41dd014e49f439f9677fe6960aa","family_id":"qf-density12-proposition-open-binding","seed":202610123031,"semantic_template":"qf-density12-proposition-open-binding/\"-7\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-2 with assignment=-9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -7)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -7)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0032","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bed20869eed2eab1722ca97d9b41b124f1aa671d299d3c7445a4dd3c7c684d7","family_id":"qf-density12-proposition-open-binding","seed":202610123032,"semantic_template":"qf-density12-proposition-open-binding/\"-6\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-1 with assignment=-7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -6)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -6)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0033","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df4d0fa5b03a38b476be85ae028845af80a9f0b408f43300f4b6bc8f865bb6b9","family_id":"qf-density12-proposition-open-binding","seed":202610123033,"semantic_template":"qf-density12-proposition-open-binding/\"-5\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤0 with assignment=-5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -5)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -5)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0034","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a0762a4037aca9d4595e9bb671f6bc22fbf962478cba5bf9c08a0db80e0f158","family_id":"qf-density12-proposition-open-binding","seed":202610123034,"semantic_template":"qf-density12-proposition-open-binding/\"-4\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤1 with assignment=-3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -4)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -4)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0035","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8db2be1ab11aad0bf9db579e14bd5897ede2fbcd43ddc5b9efcbcab6d69b312c","family_id":"qf-density12-proposition-open-binding","seed":202610123035,"semantic_template":"qf-density12-proposition-open-binding/\"-3\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤2 with assignment=-1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -3)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -3)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0036","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07c3e49009fabee8fae6f2fddc5589e04723501236e619a4c8c3faf606532fe5","family_id":"qf-density12-proposition-open-binding","seed":202610123036,"semantic_template":"qf-density12-proposition-open-binding/\"-2\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-2 with assignment=-4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -2)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -2)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0037","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ad4c881a8a9eaa9555bc03cba658b884d3ec14f215ed4e8d89baaee330a8b26","family_id":"qf-density12-proposition-open-binding","seed":202610123037,"semantic_template":"qf-density12-proposition-open-binding/\"-1\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-1 with assignment=-2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, -1)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -1)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0038","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87450bffc6da84c91ed519be356934fbba5aab40f543c58e13464d45339eaaf7","family_id":"qf-density12-proposition-open-binding","seed":202610123038,"semantic_template":"qf-density12-proposition-open-binding/\"0\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤0 with assignment=0. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, True, 0)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, 0)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0039","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8ffe6fe6658c798a22b63d7e0bcb742091dca46007449ffcc7426814f3112bc","family_id":"qf-density12-proposition-open-binding","seed":202610123039,"semantic_template":"qf-density12-proposition-open-binding/\"1\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤1 with assignment=2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 1)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 1)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0040","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc1e9757091d03035251649f69235cd5e62633dbe16f9357ee67b2700a5cc02","family_id":"qf-density12-proposition-open-binding","seed":202610123040,"semantic_template":"qf-density12-proposition-open-binding/\"2\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤2 with assignment=4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 2)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 2)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0041","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dca82e4d1d6f613309afbce5d9a5533b38c241a21602e778ff05133ec5469ad4","family_id":"qf-density12-proposition-open-binding","seed":202610123041,"semantic_template":"qf-density12-proposition-open-binding/\"3\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-2 with assignment=1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 3)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 3)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0042","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42bb6b359573918f0085c435b186ea1b476c7234bf5551784c1a65c237903413","family_id":"qf-density12-proposition-open-binding","seed":202610123042,"semantic_template":"qf-density12-proposition-open-binding/\"4\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-1 with assignment=3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 4)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 4)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0043","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d44b4272e6e7e159a4796ad96cbf4b7981049f0f6a9429cf4e3dcdf8e7ecafec","family_id":"qf-density12-proposition-open-binding","seed":202610123043,"semantic_template":"qf-density12-proposition-open-binding/\"5\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤0 with assignment=5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 5)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 5)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0044","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff13a21d1a757437329079a1ece59999c4de93882d8783a02ed3ffee9bf91b2b","family_id":"qf-density12-proposition-open-binding","seed":202610123044,"semantic_template":"qf-density12-proposition-open-binding/\"6\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤1 with assignment=7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 6)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 6)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0045","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0a0c6fadba448da1a8eb5d03c615a7e09130629f2f869b350ed7ac735ca1f95","family_id":"qf-density12-proposition-open-binding","seed":202610123045,"semantic_template":"qf-density12-proposition-open-binding/\"7\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤2 with assignment=9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 7)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 7)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0046","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecda955f644d9b796f355ee1693105f553973804aa9fe702aedc490b0d8cce9b","family_id":"qf-density12-proposition-open-binding","seed":202610123046,"semantic_template":"qf-density12-proposition-open-binding/\"8\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-2 with assignment=6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 8)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 8)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0047","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49ff2ef75d41bafd20e475ba82a2e68d49571dba87b9110fffcd9faf1a160417","family_id":"qf-density12-proposition-open-binding","seed":202610123047,"semantic_template":"qf-density12-proposition-open-binding/\"9\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤-1 with assignment=8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 9)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 9)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0048","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5e3667135bc10f4ec4798b9f539986e2d35a7c9df0fcbf2ce063b711f9d9e9a","family_id":"qf-density12-proposition-open-binding","seed":202610123048,"semantic_template":"qf-density12-proposition-open-binding/\"10\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤0 with assignment=10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 10)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 10)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0049","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00521ee8bcc80c3aa8389d2d9de55371e9c00250a04c9ddf547af057228d838","family_id":"qf-density12-proposition-open-binding","seed":202610123049,"semantic_template":"qf-density12-proposition-open-binding/\"11\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤1 with assignment=12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 11)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 11)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0050","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b2a7f181ef37f601597989c2ca5479a9b8f3f51f5d90cce85c2e31a1c498a90","family_id":"qf-density12-proposition-open-binding","seed":202610123050,"semantic_template":"qf-density12-proposition-open-binding/\"12\""},"fields":{"title":"Open-Sentence Binding and Truth Eligibility","stem":"In the classical integer context, consider x≤2 with assignment=14. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).","answer":{"kind":"expression","value":"(Yes, False, 12)","explanation":"Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 12)."},"skills":["Open-Sentence Binding and Truth Eligibility"],"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-20261008-advance-20261012-r03-proposition-0051","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16dcd8c2d9c9869b61189e182f5de67215bbf12784334784d6bf78449187b6ee","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123051,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"0\",\"-2\",\"2\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, False, False], [False, True, True], [0, -2, 2])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -2, 2])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0052","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ace577c21cc653cb6fd8b57e3f8f201a5e5ae905c3c99872cbd54dfd4c9568e","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123052,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-11\",\"-2\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [-11, -2, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-11, -2, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0053","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b32f42f637264da808524808df1078640101f12af1d7b39a2a7c6d2098db9d5","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123053,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-10\",\"-3\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-10, -3, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-10, -3, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0054","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc63a03a363a222b89663c71216880d0a704ae32d16f9505abc959721127fb4d","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123054,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-9\",\"-4\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-9, -4, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-9, -4, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0055","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cf2e66b3b8907569dc6e9201a2f36899b5f4838b6a6d4899dd1866094e3831b","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123055,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-8\",\"-5\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [-8, -5, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-8, -5, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0056","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb7581f41fc6d0d13a215327f9121d3e8cfc543a3c43e747d6879113a4a6e7a","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123056,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-7\",\"-6\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-7, -6, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-7, -6, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0057","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f5c461d505c5813e1908eecad76d45b95582956e6f4e05e191730e6d562214","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123057,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-6\",\"-7\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-6, -7, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-6, -7, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0058","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"558fb7816cb3bf926f7c9de216107d2380efe1ccf8ba5c887d5d2322ad56e422","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123058,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-5\",\"-8\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [-5, -8, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-5, -8, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0059","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a43d9e248b7d441e9a8ba9eeddf68ac2431cb5d4e1f23b0376b7d27b66618d81","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123059,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-4\",\"-9\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-4, -9, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-4, -9, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0060","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d5ae2f7a2b9b5a79a67b8480758d006f3b7d0fbeef4c8cf6a7806e33597d36","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123060,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-3\",\"-10\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-3, -10, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-3, -10, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0061","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec49c006e3081c3648024936636f57aa2a31a7b5727338774733445422f4b3a5","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123061,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-2\",\"-11\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [-2, -11, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-2, -11, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0062","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34402f04b7de8e56d891fd9099aa1b693e1e4c0ad4da7af7d530fa61362b9486","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123062,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-1\",\"-12\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [-1, -12, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-1, -12, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0063","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e5b0776aaef7d166ad66085def1e340929c7198f84dcd74c2cb12e772cf49ed","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123063,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"0\",\"-13\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, False, False], [False, True, True], [0, -13, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -13, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0064","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eda0dab6803755a3d3415a1f3ce05921de5e1019e2708b9f8d2962645bbfb6b","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123064,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"1\",\"-14\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [1, -14, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [1, -14, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0065","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"244e4a7e1548e6ce95776c01e64129eee832c99cfcc13a9dd0bc8b8f1b46ead3","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123065,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"2\",\"-15\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [2, -15, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [2, -15, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0066","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dec79fe1a0bcad0eebd83a21178b929cb64fe16d1e5987ec4495d4b88a35695","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123066,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"3\",\"-16\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [3, -16, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [3, -16, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0067","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54aa0d48a1fbbb7419e649a548bd75fd5cbc933653c15cf4e3b388dc6a41176e","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123067,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"4\",\"-17\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [4, -17, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [4, -17, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0068","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf02cc8021114cdd608dc3e2156dacf92776b84ee64071a9b091bdcae8dce7bd","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123068,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"5\",\"-18\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [5, -18, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [5, -18, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0069","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3718952adc5f9cfb441be1c3b8c47c8154048f8b800f16a538408949d2596f4","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123069,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"6\",\"-19\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [6, -19, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [6, -19, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0070","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0ec1a5eefaed9f9c30f5aa7143923d76f8aacee6fa64bcf328108a5bf5a433","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123070,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"7\",\"-20\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [7, -20, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [7, -20, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0071","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"313b0357dbdb577e376d87f063aca57d6e9bac7718f18e3e379dd026512d1a0e","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123071,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"8\",\"-21\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [8, -21, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [8, -21, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0072","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c764b6f31a67dde1da8ab5b3b3741d67cb3aa9a46fb8d8f909d27b4b1d244f4","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123072,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"9\",\"-22\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [9, -22, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [9, -22, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0073","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1e5d3e7f221842fe6da483d94aa5c3e480e1b111df187bbdb46416f833c21b2","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123073,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"10\",\"-23\",\"0\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, True], [True, True, False], [10, -23, 0])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [10, -23, 0])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0074","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09bab7b9de3efc6626bd59d82f567be811b0d78a4d7ee61e33feba9e95be2de3","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123074,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"11\",\"-24\",\"1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [11, -24, 1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [11, -24, 1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0075","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7899162638b7a4a011e24e28d551bcab47db4b9d857ea8b5ae113981c747a3e2","family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123075,"semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"12\",\"-25\",\"-1\"]"},"fields":{"title":"Equality, Negation, and Signed Difference Witnesses","stem":"In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, False, False], [True, True, True], [12, -25, -1])","explanation":"A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [12, -25, -1])."},"skills":["Equality, Negation, and Signed Difference Witnesses"],"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-20261008-advance-20261012-r03-proposition-0076","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6945b9187579b21f830753f120d7059e9da99e7dac26c3b308f7308adcf1395","family_id":"qf-density12-proposition-report-defects","seed":202610123076,"semantic_template":"qf-density12-proposition-report-defects/[[\"0\",\"-2\",\"2\"],[1,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [Yes, Yes, Yes], [0, -2, 2])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [0, -2, 2])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0077","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28387d7335838d8a9e63a9644d2540032ea1f9617b40d509607de22f2c8192f9","family_id":"qf-density12-proposition-report-defects","seed":202610123077,"semantic_template":"qf-density12-proposition-report-defects/[[\"-11\",\"-2\",\"0\"],[1,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [Yes, No, No], [-11, -2, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-11, -2, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0078","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7638a82c20c7c71ed1e0804386e13175bc9658e754cefe5d527339887c5eab1","family_id":"qf-density12-proposition-report-defects","seed":202610123078,"semantic_template":"qf-density12-proposition-report-defects/[[\"-10\",\"-3\",\"1\"],[0,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [No, Yes, Yes], [-10, -3, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, Yes, Yes], [-10, -3, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0079","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aebad644b1f7336cc54e3d9e8432586554de44f037934e03aa1de7957a9d4e9","family_id":"qf-density12-proposition-report-defects","seed":202610123079,"semantic_template":"qf-density12-proposition-report-defects/[[\"-9\",\"-4\",\"-1\"],[1,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [Yes, Yes, No], [-9, -4, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, No], [-9, -4, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0080","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dd518a69df96cbdd922c6147caa671ee13fc9857d06d888ad22d20d7877b6fb","family_id":"qf-density12-proposition-report-defects","seed":202610123080,"semantic_template":"qf-density12-proposition-report-defects/[[\"-8\",\"-5\",\"0\"],[0,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [No, No, Yes], [-8, -5, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [-8, -5, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0081","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dc4f1f6ea4e55d97d989afffc711b8811dcae8a005233311871f41c6485bb79","family_id":"qf-density12-proposition-report-defects","seed":202610123081,"semantic_template":"qf-density12-proposition-report-defects/[[\"-7\",\"-6\",\"1\"],[1,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [Yes, No, No], [-7, -6, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, No, No], [-7, -6, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0082","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4395600d2819b2b11050280fe6add71ddbc2849990023afe97bbd9919fcecad","family_id":"qf-density12-proposition-report-defects","seed":202610123082,"semantic_template":"qf-density12-proposition-report-defects/[[\"-6\",\"-7\",\"-1\"],[0,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [No, Yes, Yes], [-6, -7, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, Yes], [-6, -7, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0083","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd0fd710cd92bfd656f2f4e6daa0d7fd8137049052ea37477df8ce5819563a2","family_id":"qf-density12-proposition-report-defects","seed":202610123083,"semantic_template":"qf-density12-proposition-report-defects/[[\"-5\",\"-8\",\"0\"],[1,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [Yes, Yes, Yes], [-5, -8, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, Yes], [-5, -8, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0084","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a060b4b27db173545e840bbdfc3862e21ef081b813c4bc25070e7cf946e39ddc","family_id":"qf-density12-proposition-report-defects","seed":202610123084,"semantic_template":"qf-density12-proposition-report-defects/[[\"-4\",\"-9\",\"1\"],[0,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [No, No, Yes], [-4, -9, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, No, Yes], [-4, -9, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0085","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccde74264d231dbc6d332e2da7ee2e222af1a9a23d29e0c64b9edb0cf045b696","family_id":"qf-density12-proposition-report-defects","seed":202610123085,"semantic_template":"qf-density12-proposition-report-defects/[[\"-3\",\"-10\",\"-1\"],[1,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [Yes, No, No], [-3, -10, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-3, -10, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0086","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84c915f3843ff16016260eaeca984ce2c2be6f113dda3ff58fbceb96247b3c4","family_id":"qf-density12-proposition-report-defects","seed":202610123086,"semantic_template":"qf-density12-proposition-report-defects/[[\"-2\",\"-11\",\"0\"],[0,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [No, Yes, No], [-2, -11, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, No], [-2, -11, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0087","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6797127e42355a5a3d4f5b5ca345f2d54e4e2d1c871ecd5a4e243cfb00ef94","family_id":"qf-density12-proposition-report-defects","seed":202610123087,"semantic_template":"qf-density12-proposition-report-defects/[[\"-1\",\"-12\",\"1\"],[1,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, False], [Yes, Yes, Yes], [-1, -12, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [-1, -12, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0088","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016f355c9215d4d6a9d0f399cf8bcc847927a2bf8a08e6efea16c9feb7914f69","family_id":"qf-density12-proposition-report-defects","seed":202610123088,"semantic_template":"qf-density12-proposition-report-defects/[[\"0\",\"-13\",\"-1\"],[0,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([True, True, True], [No, No, Yes], [0, -13, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [0, -13, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0089","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b81bc4408a50156517ea516f406bd7be693c174e15fc9af7811c0ad8f5f447c","family_id":"qf-density12-proposition-report-defects","seed":202610123089,"semantic_template":"qf-density12-proposition-report-defects/[[\"1\",\"-14\",\"0\"],[1,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [No, No, Yes], [1, -14, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [1, -14, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0090","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de85f5e849112bca44d48f3ae682778cabb127fe42410d4a7417029f82656575","family_id":"qf-density12-proposition-report-defects","seed":202610123090,"semantic_template":"qf-density12-proposition-report-defects/[[\"2\",\"-15\",\"1\"],[0,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [Yes, Yes, No], [2, -15, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, Yes, No], [2, -15, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0091","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0ebffe241eb045d2c58f205e9300714ae0271d5f7cd2e05c165a4be591e4c34","family_id":"qf-density12-proposition-report-defects","seed":202610123091,"semantic_template":"qf-density12-proposition-report-defects/[[\"3\",\"-16\",\"-1\"],[1,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [No, Yes, Yes], [3, -16, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, Yes], [3, -16, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0092","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6785775eab2018e65d7fbb1d135a7e8188bb34c656dcd78482b4becaf8fd981","family_id":"qf-density12-proposition-report-defects","seed":202610123092,"semantic_template":"qf-density12-proposition-report-defects/[[\"4\",\"-17\",\"0\"],[0,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [Yes, No, No], [4, -17, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [4, -17, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0093","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c748055086d3961450aec31e96d51875f41b26ccce1668dd1d14f618bc0ddc25","family_id":"qf-density12-proposition-report-defects","seed":202610123093,"semantic_template":"qf-density12-proposition-report-defects/[[\"5\",\"-18\",\"1\"],[1,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [No, No, Yes], [5, -18, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, No, Yes], [5, -18, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0094","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e770550ad088c375df1ead91867ebc9e719d697a02f4a35e2788ce28ee9e8576","family_id":"qf-density12-proposition-report-defects","seed":202610123094,"semantic_template":"qf-density12-proposition-report-defects/[[\"6\",\"-19\",\"-1\"],[0,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [Yes, Yes, No], [6, -19, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, No], [6, -19, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0095","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6b1e1b10024e453975d96b9304455908e6c9a7a5acde3afc24c3d5b9d23c24e","family_id":"qf-density12-proposition-report-defects","seed":202610123095,"semantic_template":"qf-density12-proposition-report-defects/[[\"7\",\"-20\",\"0\"],[1,1,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [No, Yes, No], [7, -20, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, No], [7, -20, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0096","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c38f4194ba64298545182c3e5a77f5c90c303fdb57c0ea0a75cafac6447298b8","family_id":"qf-density12-proposition-report-defects","seed":202610123096,"semantic_template":"qf-density12-proposition-report-defects/[[\"8\",\"-21\",\"1\"],[0,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [Yes, No, No], [8, -21, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, No, No], [8, -21, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0097","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"209c13e87beeee699f99e2188d3d355fbb272bcfecb144f000fa45977e840f68","family_id":"qf-density12-proposition-report-defects","seed":202610123097,"semantic_template":"qf-density12-proposition-report-defects/[[\"9\",\"-22\",\"-1\"],[1,0,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [No, No, Yes], [9, -22, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [9, -22, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0098","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af70a5382d0c416d7719916d3de9ac9eda15a2c5d711fc4346dc32d8028ead19","family_id":"qf-density12-proposition-report-defects","seed":202610123098,"semantic_template":"qf-density12-proposition-report-defects/[[\"10\",\"-23\",\"0\"],[0,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [Yes, Yes, Yes], [10, -23, 0])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, Yes], [10, -23, 0])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0099","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05e2e93b07c2e9f98079acc17ac66bc4f53a88267d126a27b7066f7f20496f3a","family_id":"qf-density12-proposition-report-defects","seed":202610123099,"semantic_template":"qf-density12-proposition-report-defects/[[\"11\",\"-24\",\"1\"],[1,1,1]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, False], [No, Yes, No], [11, -24, 1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, Yes, No], [11, -24, 1])."},"skills":["Candidate Truth Reports Checked Independently"],"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-20261008-advance-20261012-r03-proposition-0100","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c96eacaca10b832a57e948c013c63e807bedad3aae6b099e73ae2ebd03d3b23d","family_id":"qf-density12-proposition-report-defects","seed":202610123100,"semantic_template":"qf-density12-proposition-report-defects/[[\"12\",\"-25\",\"-1\"],[0,0,0]]"},"fields":{"title":"Candidate Truth Reports Checked Independently","stem":"In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).","answer":{"kind":"expression","value":"([False, True, True], [Yes, No, No], [12, -25, -1])","explanation":"Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [12, -25, -1])."},"skills":["Candidate Truth Reports Checked Independently"],"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."}}