{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0001","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fc5edfe16042927a001de0519f39cbc952594901011f9431b423a1b37984437","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197001,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P0\",\"context\":[{\"name\":\"h0\",\"type\":\"P0\"}],\"term\":{\"kind\":\"var\",\"name\":\"h0\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h0\",\"type\":\"P0\"}], term={\"kind\":\"var\",\"name\":\"h0\"}, claimed_type=\"P0\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P0, P0, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P0, P0, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0002","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40a72cb72042396bed656012cddccbfca4149beb57e96388e40628572671fe10","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197002,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P1\",\"P1\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x1\"},\"kind\":\"lam\",\"name\":\"x1\",\"var_type\":\"P1\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x1\",\"var_type\":\"P1\",\"body\":{\"kind\":\"var\",\"name\":\"x1\"}}, claimed_type=[\"imp\",\"P1\",\"P1\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P1, P1], [imp, P1, P1], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P1, P1], [imp, P1, P1], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0003","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff9ad0fb8893e8f0d0a94649b005cd1f50bf93e45cefece38dbd974b6e1c6d7","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197003,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P2\",\"Q2\"],\"context\":[{\"name\":\"h2\",\"type\":\"P2\"},{\"name\":\"q2\",\"type\":\"Q2\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h2\"},\"right\":{\"kind\":\"var\",\"name\":\"q2\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h2\",\"type\":\"P2\"},{\"name\":\"q2\",\"type\":\"Q2\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h2\"},\"right\":{\"kind\":\"var\",\"name\":\"q2\"}}, claimed_type=[\"and\",\"P2\",\"Q2\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P2, Q2], [and, P2, Q2], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P2, Q2], [and, P2, Q2], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0004","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b46bb43060211f68a774e95cd43f1dcde6d462aebbbcf817fa712e446f4d3598","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197004,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P3\",\"context\":[{\"name\":\"pair3\",\"type\":[\"and\",\"P3\",\"Q3\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair3\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair3\",\"type\":[\"and\",\"P3\",\"Q3\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair3\"}}, claimed_type=\"P3\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P3, P3, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P3, P3, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0005","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50613c308bc1e484acc56b5c227e96931f09406a9209f1c6973209f8d7af2c2a","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197005,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P4\",\"context\":[{\"name\":\"h4\",\"type\":\"P4\"}],\"term\":{\"kind\":\"var\",\"name\":\"h4\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h4\",\"type\":\"P4\"}], term={\"kind\":\"var\",\"name\":\"h4\"}, claimed_type=\"P4\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P4, P4, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P4, P4, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0006","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ab814593730f45bb9c3f3790487908fe1674401d2cee6971268947ef25cbee","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197006,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P5\",\"P5\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x5\"},\"kind\":\"lam\",\"name\":\"x5\",\"var_type\":\"P5\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x5\",\"var_type\":\"P5\",\"body\":{\"kind\":\"var\",\"name\":\"x5\"}}, claimed_type=[\"imp\",\"P5\",\"P5\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P5, P5], [imp, P5, P5], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P5, P5], [imp, P5, P5], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0007","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"784c3a1f8211b83fea79fe047fd0c0fe26bd811e6dda205559016c6e9af0e568","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197007,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P6\",\"Q6\"],\"context\":[{\"name\":\"h6\",\"type\":\"P6\"},{\"name\":\"q6\",\"type\":\"Q6\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h6\"},\"right\":{\"kind\":\"var\",\"name\":\"q6\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h6\",\"type\":\"P6\"},{\"name\":\"q6\",\"type\":\"Q6\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h6\"},\"right\":{\"kind\":\"var\",\"name\":\"q6\"}}, claimed_type=[\"and\",\"P6\",\"Q6\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P6, Q6], [and, P6, Q6], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P6, Q6], [and, P6, Q6], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0008","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de58b69251b4b09aebfdf41927c202ff9eb44a5ea3f4108b35abdc484a24b97b","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197008,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P7\",\"context\":[{\"name\":\"pair7\",\"type\":[\"and\",\"P7\",\"Q7\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair7\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair7\",\"type\":[\"and\",\"P7\",\"Q7\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair7\"}}, claimed_type=\"P7\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P7, P7, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P7, P7, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0009","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e73052071947382e0ad142d2ef52f1b6cba34a8a81bb000e49d3f94f93c3050f","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197009,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P8\",\"context\":[{\"name\":\"h8\",\"type\":\"P8\"}],\"term\":{\"kind\":\"var\",\"name\":\"h8\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h8\",\"type\":\"P8\"}], term={\"kind\":\"var\",\"name\":\"h8\"}, claimed_type=\"P8\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P8, P8, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P8, P8, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0010","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d0c096bfe66ff6678409d2f7553aa7be6548fc17a939d31bcda7e598323f55","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197010,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P9\",\"P9\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x9\"},\"kind\":\"lam\",\"name\":\"x9\",\"var_type\":\"P9\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x9\",\"var_type\":\"P9\",\"body\":{\"kind\":\"var\",\"name\":\"x9\"}}, claimed_type=[\"imp\",\"P9\",\"P9\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P9, P9], [imp, P9, P9], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P9, P9], [imp, P9, P9], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0011","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12f0d14da945417b89ef1fa215ffd4af2acf2198b72557bd60acfb54315dd689","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197011,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P10\",\"Q10\"],\"context\":[{\"name\":\"h10\",\"type\":\"P10\"},{\"name\":\"q10\",\"type\":\"Q10\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h10\"},\"right\":{\"kind\":\"var\",\"name\":\"q10\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h10\",\"type\":\"P10\"},{\"name\":\"q10\",\"type\":\"Q10\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h10\"},\"right\":{\"kind\":\"var\",\"name\":\"q10\"}}, claimed_type=[\"and\",\"P10\",\"Q10\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P10, Q10], [and, P10, Q10], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P10, Q10], [and, P10, Q10], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0012","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b1453824dc7218882d3a4fab37e1152bad5e562f1550b434ee38d114398dd78","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197012,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P11\",\"context\":[{\"name\":\"pair11\",\"type\":[\"and\",\"P11\",\"Q11\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair11\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair11\",\"type\":[\"and\",\"P11\",\"Q11\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair11\"}}, claimed_type=\"P11\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P11, P11, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P11, P11, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0013","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95b2bf76c828f6699f6fa3ef67844e5d8054268800d5388544f154a250cd6d22","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197013,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P12\",\"context\":[{\"name\":\"h12\",\"type\":\"P12\"}],\"term\":{\"kind\":\"var\",\"name\":\"h12\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h12\",\"type\":\"P12\"}], term={\"kind\":\"var\",\"name\":\"h12\"}, claimed_type=\"P12\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P12, P12, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P12, P12, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0014","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aab17dae22708fa29f90b64f5069ca3efe6204a03533914a7abf06a9d5ed7352","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197014,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P13\",\"P13\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x13\"},\"kind\":\"lam\",\"name\":\"x13\",\"var_type\":\"P13\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x13\",\"var_type\":\"P13\",\"body\":{\"kind\":\"var\",\"name\":\"x13\"}}, claimed_type=[\"imp\",\"P13\",\"P13\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P13, P13], [imp, P13, P13], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P13, P13], [imp, P13, P13], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0015","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e142a29a9440af4a228acdcbe98a4c8e889aa29f3edc2cbd80bbf7d138980f4e","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197015,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P14\",\"Q14\"],\"context\":[{\"name\":\"h14\",\"type\":\"P14\"},{\"name\":\"q14\",\"type\":\"Q14\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h14\"},\"right\":{\"kind\":\"var\",\"name\":\"q14\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h14\",\"type\":\"P14\"},{\"name\":\"q14\",\"type\":\"Q14\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h14\"},\"right\":{\"kind\":\"var\",\"name\":\"q14\"}}, claimed_type=[\"and\",\"P14\",\"Q14\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P14, Q14], [and, P14, Q14], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P14, Q14], [and, P14, Q14], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0016","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"630385bf6bb58e1b16fa69f82a3a9888ed65c5f3b03fb249867aaeb2222aa1af","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197016,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P15\",\"context\":[{\"name\":\"pair15\",\"type\":[\"and\",\"P15\",\"Q15\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair15\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair15\",\"type\":[\"and\",\"P15\",\"Q15\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair15\"}}, claimed_type=\"P15\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P15, P15, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P15, P15, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0017","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8a4b20f3960b777eaa75306e30af909a476dbbe83740cb1091092f94271c7ee","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197017,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P16\",\"context\":[{\"name\":\"h16\",\"type\":\"P16\"}],\"term\":{\"kind\":\"var\",\"name\":\"h16\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h16\",\"type\":\"P16\"}], term={\"kind\":\"var\",\"name\":\"h16\"}, claimed_type=\"P16\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P16, P16, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P16, P16, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0018","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42734e1fc1506cd68377aa7a724336483d004df61b3fe5a67747de1974580fa7","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197018,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P17\",\"P17\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x17\"},\"kind\":\"lam\",\"name\":\"x17\",\"var_type\":\"P17\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x17\",\"var_type\":\"P17\",\"body\":{\"kind\":\"var\",\"name\":\"x17\"}}, claimed_type=[\"imp\",\"P17\",\"P17\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P17, P17], [imp, P17, P17], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P17, P17], [imp, P17, P17], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0019","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f8a3ab2b1b9e73fb5333efe5d1b78d7a79c17d112b4a187d991efb01889c67a","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197019,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P18\",\"Q18\"],\"context\":[{\"name\":\"h18\",\"type\":\"P18\"},{\"name\":\"q18\",\"type\":\"Q18\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h18\"},\"right\":{\"kind\":\"var\",\"name\":\"q18\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h18\",\"type\":\"P18\"},{\"name\":\"q18\",\"type\":\"Q18\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h18\"},\"right\":{\"kind\":\"var\",\"name\":\"q18\"}}, claimed_type=[\"and\",\"P18\",\"Q18\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P18, Q18], [and, P18, Q18], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P18, Q18], [and, P18, Q18], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0020","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15eb30e56ae73ba933f5ccacff99565341cedf8d9e501b524fc2c5f837a14762","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197020,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P19\",\"context\":[{\"name\":\"pair19\",\"type\":[\"and\",\"P19\",\"Q19\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair19\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair19\",\"type\":[\"and\",\"P19\",\"Q19\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair19\"}}, claimed_type=\"P19\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P19, P19, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P19, P19, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0021","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9151640d9651d6741e6aeb5ad98768cce811618c6a4015d10dc7ce5013a4cced","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197021,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P20\",\"context\":[{\"name\":\"h20\",\"type\":\"P20\"}],\"term\":{\"kind\":\"var\",\"name\":\"h20\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h20\",\"type\":\"P20\"}], term={\"kind\":\"var\",\"name\":\"h20\"}, claimed_type=\"P20\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P20, P20, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P20, P20, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0022","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b5fdd9352c4eea2692c4773a98c620b2cd425f4e79fb5e2deba3a95130ac0bd","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197022,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"imp\",\"P21\",\"P21\"],\"context\":[],\"term\":{\"body\":{\"kind\":\"var\",\"name\":\"x21\"},\"kind\":\"lam\",\"name\":\"x21\",\"var_type\":\"P21\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={\"kind\":\"lam\",\"name\":\"x21\",\"var_type\":\"P21\",\"body\":{\"kind\":\"var\",\"name\":\"x21\"}}, claimed_type=[\"imp\",\"P21\",\"P21\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([imp, P21, P21], [imp, P21, P21], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P21, P21], [imp, P21, P21], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0023","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a2a7d32ef79b6cb7384b44d4848b8d73f87b56ff797b6aca76b74058c9d9ba","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197023,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":[\"and\",\"P22\",\"Q22\"],\"context\":[{\"name\":\"h22\",\"type\":\"P22\"},{\"name\":\"q22\",\"type\":\"Q22\"}],\"term\":{\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h22\"},\"right\":{\"kind\":\"var\",\"name\":\"q22\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h22\",\"type\":\"P22\"},{\"name\":\"q22\",\"type\":\"Q22\"}], term={\"kind\":\"pair\",\"left\":{\"kind\":\"var\",\"name\":\"h22\"},\"right\":{\"kind\":\"var\",\"name\":\"q22\"}}, claimed_type=[\"and\",\"P22\",\"Q22\"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"([and, P22, Q22], [and, P22, Q22], Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P22, Q22], [and, P22, Q22], Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0024","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c32cdcccbf7900f81bf711f762f27bf2e3ea614690bdab2797e3cb2b7008febf","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197024,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P23\",\"context\":[{\"name\":\"pair23\",\"type\":[\"and\",\"P23\",\"Q23\"]}],\"term\":{\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair23\"}}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"pair23\",\"type\":[\"and\",\"P23\",\"Q23\"]}], term={\"kind\":\"fst\",\"pair\":{\"kind\":\"var\",\"name\":\"pair23\"}}, claimed_type=\"P23\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P23, P23, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P23, P23, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0025","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"342ad0ad7ad16fb7e06cc2bf912d99757bd3a12cdfbfe14b0db53b687873cae2","family_id":"qf-ext19-proofterm-small-calculus","seed":202610197025,"semantic_template":"qf-ext19-proofterm-small-calculus/{\"claimed_type\":\"P24\",\"context\":[{\"name\":\"h24\",\"type\":\"P24\"}],\"term\":{\"kind\":\"var\",\"name\":\"h24\"}}"},"fields":{"title":"Synthetic Implication/Conjunction Term Type against a Claim","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h24\",\"type\":\"P24\"}], term={\"kind\":\"var\",\"name\":\"h24\"}, claimed_type=\"P24\". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).","answer":{"kind":"expression","value":"(P24, P24, Yes)","explanation":"Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P24, P24, Yes)."},"skills":["Synthetic Implication/Conjunction Term Type against a Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0026","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"621a21c4cb16665696c1c01ce374a034e3fe648f58f1ceb4ebf2f4d39cfa11c9","family_id":"qf-ext19-proofterm-context-use","seed":202610197026,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h0\",\"type\":\"P0\"}],\"term\":{\"kind\":\"var\",\"name\":\"h0\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h0\",\"type\":\"P0\"}], term={\"kind\":\"var\",\"name\":\"h0\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h0], [h0], [], P0, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h0], [h0], [], P0, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0027","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22180bc4b2f829827aeedb64f87dfd3665866e45f456f715eaaa1c9545712dbd","family_id":"qf-ext19-proofterm-context-use","seed":202610197027,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h1\",\"type\":\"P1\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing1\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h1\",\"type\":\"P1\"}], term={\"kind\":\"var\",\"name\":\"missing1\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing1], [h1], [missing1], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing1], [h1], [missing1], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0028","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f53d91e665563f3e436a6434e3edf65519e40e994485b987bb4fd256297b6b87","family_id":"qf-ext19-proofterm-context-use","seed":202610197028,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h2\",\"type\":\"P2\"}],\"term\":{\"kind\":\"var\",\"name\":\"h2\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h2\",\"type\":\"P2\"}], term={\"kind\":\"var\",\"name\":\"h2\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h2], [h2], [], P2, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h2], [h2], [], P2, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0029","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0219b3589d8212320687838f73868a55a155a55bc7f4fb0bb39a4181f5feb15d","family_id":"qf-ext19-proofterm-context-use","seed":202610197029,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h3\",\"type\":\"P3\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing3\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h3\",\"type\":\"P3\"}], term={\"kind\":\"var\",\"name\":\"missing3\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing3], [h3], [missing3], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing3], [h3], [missing3], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0030","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08c39d11ecf967563043af6c3e55681ee40679836882d0d7cd5f064c958be3a8","family_id":"qf-ext19-proofterm-context-use","seed":202610197030,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h4\",\"type\":\"P4\"}],\"term\":{\"kind\":\"var\",\"name\":\"h4\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h4\",\"type\":\"P4\"}], term={\"kind\":\"var\",\"name\":\"h4\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h4], [h4], [], P4, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h4], [h4], [], P4, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0031","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0b3552997462ac9a9b4ca5450ac8365d920b86d9bedb504d9742f7d51aaac4e","family_id":"qf-ext19-proofterm-context-use","seed":202610197031,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h5\",\"type\":\"P5\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing5\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h5\",\"type\":\"P5\"}], term={\"kind\":\"var\",\"name\":\"missing5\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing5], [h5], [missing5], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing5], [h5], [missing5], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0032","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87051255f9853baf3b67c6058358fec330fd63c4fa4c67f0149e181992699c2c","family_id":"qf-ext19-proofterm-context-use","seed":202610197032,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h6\",\"type\":\"P6\"}],\"term\":{\"kind\":\"var\",\"name\":\"h6\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h6\",\"type\":\"P6\"}], term={\"kind\":\"var\",\"name\":\"h6\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h6], [h6], [], P6, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h6], [h6], [], P6, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0033","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93e2861bb5b15f3c9afa8b5ae701d6a9c6a9d145823015ef3d4c7e5e840d1dc","family_id":"qf-ext19-proofterm-context-use","seed":202610197033,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h7\",\"type\":\"P7\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing7\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h7\",\"type\":\"P7\"}], term={\"kind\":\"var\",\"name\":\"missing7\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing7], [h7], [missing7], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing7], [h7], [missing7], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0034","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a8390c36ddabb6385a5de10a344b41ec24acdabe6e3c7692723e279fd442072","family_id":"qf-ext19-proofterm-context-use","seed":202610197034,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h8\",\"type\":\"P8\"}],\"term\":{\"kind\":\"var\",\"name\":\"h8\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h8\",\"type\":\"P8\"}], term={\"kind\":\"var\",\"name\":\"h8\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h8], [h8], [], P8, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h8], [h8], [], P8, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0035","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1ef2de8c10226ddc8e7c59ae7adfdcd0c3ebc3b255679424cde529aacf15dcb","family_id":"qf-ext19-proofterm-context-use","seed":202610197035,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h9\",\"type\":\"P9\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing9\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h9\",\"type\":\"P9\"}], term={\"kind\":\"var\",\"name\":\"missing9\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing9], [h9], [missing9], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing9], [h9], [missing9], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0036","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57bd562c8126ddbd6d808273f1055953ef7ec8dd211bd4edb347116464597e69","family_id":"qf-ext19-proofterm-context-use","seed":202610197036,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h10\",\"type\":\"P10\"}],\"term\":{\"kind\":\"var\",\"name\":\"h10\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h10\",\"type\":\"P10\"}], term={\"kind\":\"var\",\"name\":\"h10\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h10], [h10], [], P10, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h10], [h10], [], P10, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0037","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ecdec70f3c7abf0a558b59c8ce003185dea34c21e5cbf3fb442330921524764","family_id":"qf-ext19-proofterm-context-use","seed":202610197037,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h11\",\"type\":\"P11\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing11\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h11\",\"type\":\"P11\"}], term={\"kind\":\"var\",\"name\":\"missing11\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing11], [h11], [missing11], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing11], [h11], [missing11], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0038","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ab3969c0e16bf8a4371d68ab768e4488489cb05ba8ecfd1825379bd5b39eee","family_id":"qf-ext19-proofterm-context-use","seed":202610197038,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h12\",\"type\":\"P12\"}],\"term\":{\"kind\":\"var\",\"name\":\"h12\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h12\",\"type\":\"P12\"}], term={\"kind\":\"var\",\"name\":\"h12\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h12], [h12], [], P12, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h12], [h12], [], P12, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0039","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"310185944942b601bea000e95866107970c420120d0388f8f6d470b8762783b0","family_id":"qf-ext19-proofterm-context-use","seed":202610197039,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h13\",\"type\":\"P13\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing13\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h13\",\"type\":\"P13\"}], term={\"kind\":\"var\",\"name\":\"missing13\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing13], [h13], [missing13], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing13], [h13], [missing13], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0040","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b52da623570959d513049082db05dafa8c5fee72b5b6721eaf5e8d182d270a5","family_id":"qf-ext19-proofterm-context-use","seed":202610197040,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h14\",\"type\":\"P14\"}],\"term\":{\"kind\":\"var\",\"name\":\"h14\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h14\",\"type\":\"P14\"}], term={\"kind\":\"var\",\"name\":\"h14\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h14], [h14], [], P14, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h14], [h14], [], P14, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0041","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f3ebf66a0c43098b15346308df5f805abaa97c2dd5e76e130ec3bac0ce542e7","family_id":"qf-ext19-proofterm-context-use","seed":202610197041,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h15\",\"type\":\"P15\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing15\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h15\",\"type\":\"P15\"}], term={\"kind\":\"var\",\"name\":\"missing15\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing15], [h15], [missing15], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing15], [h15], [missing15], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0042","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aed9b2fb6e17544a0e4eb2e1f5cf5d5861dc94deecb7cac5ec09d41a3c2df6b","family_id":"qf-ext19-proofterm-context-use","seed":202610197042,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h16\",\"type\":\"P16\"}],\"term\":{\"kind\":\"var\",\"name\":\"h16\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h16\",\"type\":\"P16\"}], term={\"kind\":\"var\",\"name\":\"h16\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h16], [h16], [], P16, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h16], [h16], [], P16, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0043","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4273a2aac3ef6566f17a7e8279b7ffb237c0e1e169705edc3406b3acce69460d","family_id":"qf-ext19-proofterm-context-use","seed":202610197043,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h17\",\"type\":\"P17\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing17\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h17\",\"type\":\"P17\"}], term={\"kind\":\"var\",\"name\":\"missing17\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing17], [h17], [missing17], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing17], [h17], [missing17], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0044","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e7746f43d4b0997fffc41fc6bdf401ad2172a7256b36e64b4acc3edf0497791","family_id":"qf-ext19-proofterm-context-use","seed":202610197044,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h18\",\"type\":\"P18\"}],\"term\":{\"kind\":\"var\",\"name\":\"h18\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h18\",\"type\":\"P18\"}], term={\"kind\":\"var\",\"name\":\"h18\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h18], [h18], [], P18, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h18], [h18], [], P18, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0045","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ac0a9db0158a456e7f04bfd65fc0f6c807ca4675d664da0befd91f3ecfc551","family_id":"qf-ext19-proofterm-context-use","seed":202610197045,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h19\",\"type\":\"P19\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing19\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h19\",\"type\":\"P19\"}], term={\"kind\":\"var\",\"name\":\"missing19\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing19], [h19], [missing19], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing19], [h19], [missing19], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0046","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c405d84c13f636540d8de1075586b84897f114f52f5ec4fc8245237eb65060d","family_id":"qf-ext19-proofterm-context-use","seed":202610197046,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h20\",\"type\":\"P20\"}],\"term\":{\"kind\":\"var\",\"name\":\"h20\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h20\",\"type\":\"P20\"}], term={\"kind\":\"var\",\"name\":\"h20\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h20], [h20], [], P20, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h20], [h20], [], P20, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0047","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"172ec88cefad71fa68215b2992bfb49d85f2902f0b6b07be0f661bda4f4388ad","family_id":"qf-ext19-proofterm-context-use","seed":202610197047,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h21\",\"type\":\"P21\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing21\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h21\",\"type\":\"P21\"}], term={\"kind\":\"var\",\"name\":\"missing21\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing21], [h21], [missing21], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing21], [h21], [missing21], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0048","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cc80498d810f679679f2c31a487a4a7adc6a89fda37baedd73327f490fd5776","family_id":"qf-ext19-proofterm-context-use","seed":202610197048,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h22\",\"type\":\"P22\"}],\"term\":{\"kind\":\"var\",\"name\":\"h22\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h22\",\"type\":\"P22\"}], term={\"kind\":\"var\",\"name\":\"h22\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h22], [h22], [], P22, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h22], [h22], [], P22, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0049","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6926671a7d21ed3ea5b7d11183d768a6a4c1397c1d73946605eba5b3be406beb","family_id":"qf-ext19-proofterm-context-use","seed":202610197049,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h23\",\"type\":\"P23\"}],\"term\":{\"kind\":\"var\",\"name\":\"missing23\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h23\",\"type\":\"P23\"}], term={\"kind\":\"var\",\"name\":\"missing23\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([missing23], [h23], [missing23], not-evaluated, No)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing23], [h23], [missing23], not-evaluated, No)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0050","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"436eb6afb68b09968d08aa5d6d81b0849a6e0296015b209a175d247ce37a58c2","family_id":"qf-ext19-proofterm-context-use","seed":202610197050,"semantic_template":"qf-ext19-proofterm-context-use/{\"context\":[{\"name\":\"h24\",\"type\":\"P24\"}],\"term\":{\"kind\":\"var\",\"name\":\"h24\"}}"},"fields":{"title":"Synthetic Free Variables before Type Inference","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{\"name\":\"h24\",\"type\":\"P24\"}], term={\"kind\":\"var\",\"name\":\"h24\"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.","answer":{"kind":"expression","value":"([h24], [h24], [], P24, Yes)","explanation":"Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h24], [h24], [], P24, Yes)."},"skills":["Synthetic Free Variables before Type Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0051","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbdffdd98ff5d9343da462fc0206112e49b351e5530458e11e6ec00cc7456f8a","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197051,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":false,\"tactic_finished\":false,\"term_present\":false,\"unresolved_goals\":0}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=false, term_present=false, unresolved_goals=0. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(No, No, No, 0, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, No, No, 0, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0052","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca261d154470b6d8f42f833c8de1161d36c778ee2ca8abc2d7d120a02e92730","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197052,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":1}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=1. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 1, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 1, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0053","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df6635d8f2bf4401919e67d720341af5223c6708705b58edf132b2f2022ef250","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197053,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":2}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=2. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 2, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 2, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0054","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ecafe5ddfa3fc629e5d87af792a18c50bb46f1fe0aea8a35933034f42d4b58a","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197054,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":3}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=3. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, No, 3, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 3, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0055","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eb4e67286324aad8be487f9750f53da90f8911b961e1e508fa419d8093e9b43","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197055,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":false,\"term_present\":true,\"unresolved_goals\":4}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=4. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, No, Yes, 4, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 4, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0056","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae5740e12b216e7ffb1b65bc2070d5e292bde1d96611f00a8a18cce314cc3d49","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197056,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":false,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":5}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=true, unresolved_goals=5. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(No, Yes, Yes, 5, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, Yes, 5, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0057","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8873cc071e6ca0193167754ff18c368caffb10d9e1ab626b837b3ac8d6b7bb70","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197057,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":6}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=6. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, No, 6, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 6, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0058","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b86bcb6563436edfe449dda53f2127541155ff97d69e919f106ac652aaaad0d","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197058,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":7}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=7. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 7, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 7, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0059","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bbe75b038a862e9a1ecca77ccc473e322ab32e4565bb2d8e0d0487dc6508ba9","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197059,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":false,\"term_present\":true,\"unresolved_goals\":8}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=8. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, No, Yes, 8, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 8, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0060","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b250d5e9468970a242aa67794ac6788e56d94db94cec399e29d6e75461895bd","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197060,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":9}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=9. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, No, 9, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 9, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0061","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63269c4c5abbc5842fb67fe9255d399e7e973ac198e2e5f38ae045da0b42c3c0","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197061,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":false,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":10}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=true, unresolved_goals=10. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(No, Yes, Yes, 10, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, Yes, 10, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0062","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c0ca8f17ffcd8e1904f5aa22e65abc3d17691c2997a8353e0f4f5ad10ce560","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197062,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":11}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=11. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 11, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 11, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0063","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f6272fe50b3fe395c44cb888afa6f8de37323953d9164ecb6c882bc103d8dd2","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197063,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":false,\"term_present\":false,\"unresolved_goals\":12}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=false, unresolved_goals=12. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, No, No, 12, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, No, 12, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0064","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4a61dc34d2a6794794c711e32cc2ec580147288937d08d058dc31117ebfd8b2","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197064,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":13}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=13. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 13, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 13, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0065","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d26f5f552625b23fc92700d4d99819e7b47994f46d9b294b0841ef948a207809","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197065,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":14}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=14. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 14, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 14, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0066","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f32ce6a723d624231240167d04f300b542da5072bc6de1156160d5198f30b21","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197066,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":false,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":15}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=false, unresolved_goals=15. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(No, Yes, No, 15, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, No, 15, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0067","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d9dde2a4e236eec29da66d644be6fa5b433a8393b6eb1b3840399dcafe4bc48","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197067,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":false,\"term_present\":true,\"unresolved_goals\":16}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=16. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, No, Yes, 16, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 16, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0068","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27157382803a6e3420f63642c1974fd83ad3d3891ef6cd98a7957a0a24316b2c","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197068,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":17}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=17. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 17, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 17, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0069","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dbcbd314769bf5cfce01b41b9bfa1f3c996a8ba1f0b06fb503c3b38af7f084b","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197069,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":18}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=18. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, No, 18, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 18, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0070","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d16be081a19080a479e92feff0a449654ecd9d2eee82cde7e414d4e78410ef","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197070,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":19}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=19. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 19, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 19, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0071","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0301b6ed2badd7bfcb7a561b84642a59507df6318812a8736ba55d7059408a8","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197071,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":false,\"tactic_finished\":false,\"term_present\":true,\"unresolved_goals\":20}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=false, term_present=true, unresolved_goals=20. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(No, No, Yes, 20, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, No, Yes, 20, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0072","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e8df3db0425b26c0884dea46697772575358e77b6be9ca7c88ea0d9022cfc0","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197072,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":false,\"unresolved_goals\":21}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=21. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, No, 21, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 21, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0073","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c53cc77f07ebf3d2433407a3cb7288b7d37d6dc52f2a122d2e683f5b110e568f","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197073,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":22}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=22. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 22, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 22, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0074","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa900428cf43f7537e2029c3c42e1b7c275de6daf42884e836365dd693d326f9","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197074,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":true,\"term_present\":true,\"unresolved_goals\":23}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=23. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, Yes, Yes, 23, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 23, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0075","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de63a75b5cbf16b0b2d9bcde9b90995d69e45ff05dbd9395aa10e3a6955e703e","family_id":"qf-ext19-proofterm-tactic-vs-term","seed":202610197075,"semantic_template":"qf-ext19-proofterm-tactic-vs-term/{\"statement_ready\":true,\"tactic_finished\":false,\"term_present\":false,\"unresolved_goals\":24}"},"fields":{"title":"Synthetic Tactic Completion Is Not a Kernel Receipt","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=false, unresolved_goals=24. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.","answer":{"kind":"expression","value":"(Yes, No, No, 24, No, No, No, no-kernel receipt)","explanation":"A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, No, 24, No, No, No, no-kernel receipt)."},"skills":["Synthetic Tactic Completion Is Not a Kernel Receipt"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0076","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c092ee6eb45d5021e1261c8a5fa29b2b4d4cbf4f097cdfa27f8b390ffa4a4af4","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197076,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":false,\"environment_id\":\"\",\"kernel_checked\":true,\"statement_id\":\"\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"\",\"statement_id\":\"\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\",\"kernel_checked\":true,\"axioms_recorded\":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(missing, missing, No, No, Yes, No, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, missing, No, No, Yes, No, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0077","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45b85ec58bf623e5f14156eacda7ace6f8441d9b4f77e1b7a3d6fa6c84fe8fc6","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197077,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-1\",\"kernel_checked\":false,\"statement_id\":\"thm-1\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000001\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000065\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-1\",\"statement_id\":\"thm-1\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000001\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000065\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-1, thm-1, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-1, thm-1, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0078","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc0888494048392cf6d7a4576c61d4eda84442520c86869b30b96b58d2067dc","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197078,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-2\",\"kernel_checked\":true,\"statement_id\":\"thm-2\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000002\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000066\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-2\",\"statement_id\":\"thm-2\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000002\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000066\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-2, thm-2, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-2, thm-2, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0079","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d78269482032f1c7d46e90702daa9dd41b0f1aa41dd8e84ed77e4150047e6de4","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197079,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-3\",\"kernel_checked\":false,\"statement_id\":\"thm-3\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000003\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-3\",\"statement_id\":\"thm-3\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000003\",\"type_sha256\":\"bad\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-3, thm-3, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-3, thm-3, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0080","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13a523c2d270266b38be29e23ac7a04fb6f844727526f1e30628e11caf63f997","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197080,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-4\",\"kernel_checked\":true,\"statement_id\":\"thm-4\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000068\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-4\",\"statement_id\":\"thm-4\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000068\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-4, thm-4, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-4, thm-4, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0081","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d48f18e75a7ba1026a5ddbf926de0f66d63941d16797d71f335132684cf9b59","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197081,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-5\",\"kernel_checked\":false,\"statement_id\":\"\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000005\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000069\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-5\",\"statement_id\":\"\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000005\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000069\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-5, missing, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-5, missing, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0082","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39bc915f7e5e2928a392c241bfb9e7a30cc0ffa92ec9bde34aa3a174bab355ad","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197082,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"\",\"kernel_checked\":true,\"statement_id\":\"thm-6\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000006\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"\",\"statement_id\":\"thm-6\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000006\",\"type_sha256\":\"bad\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(missing, thm-6, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-6, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0083","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9dffdda1ec968e441ef4841d9c95fa855416727158b266725e378c502f8bdb5","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197083,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":false,\"environment_id\":\"env-7\",\"kernel_checked\":false,\"statement_id\":\"thm-7\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000007\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006b\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-7\",\"statement_id\":\"thm-7\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000007\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006b\",\"kernel_checked\":false,\"axioms_recorded\":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-7, thm-7, Yes, Yes, No, No, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-7, thm-7, Yes, Yes, No, No, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0084","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ec218a04801094441c69ad5580b6d1d5a94d15497ea2b5f48921296813866c","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197084,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-8\",\"kernel_checked\":true,\"statement_id\":\"thm-8\",\"term_sha256\":\"bad\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006c\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-8\",\"statement_id\":\"thm-8\",\"term_sha256\":\"bad\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006c\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-8, thm-8, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-8, thm-8, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0085","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac912789f762c63c502ab9ce99b8fbc24b61aff39b2c9984d99ebd539f52f9c","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197085,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-9\",\"kernel_checked\":false,\"statement_id\":\"thm-9\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000009\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-9\",\"statement_id\":\"thm-9\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000009\",\"type_sha256\":\"bad\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-9, thm-9, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-9, thm-9, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0086","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c76c1ec66f6e25e19661c9360754dd73896adf2f9d7c31f65a1f128068bf81d","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197086,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-10\",\"kernel_checked\":true,\"statement_id\":\"\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000a\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006e\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-10\",\"statement_id\":\"\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000a\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006e\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-10, missing, Yes, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-10, missing, Yes, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0087","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96b58e8d7d8c570022e86400e0c10ee5982bad2513e941e209c49f548069ea7e","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197087,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-11\",\"kernel_checked\":false,\"statement_id\":\"thm-11\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000b\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006f\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-11\",\"statement_id\":\"thm-11\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000b\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000006f\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-11, thm-11, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-11, thm-11, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0088","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9923a61a78c5544ddc770cfe86ff4ae841f81c5278df24be99547bdad84bd9dc","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197088,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"\",\"kernel_checked\":true,\"statement_id\":\"thm-12\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"\",\"statement_id\":\"thm-12\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(missing, thm-12, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-12, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0089","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dbce0cbd9699c0926736439d4cf8acb2a22f641f9b6f9ab43833104577493fd","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197089,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-13\",\"kernel_checked\":false,\"statement_id\":\"thm-13\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000d\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000071\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-13\",\"statement_id\":\"thm-13\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000d\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000071\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-13, thm-13, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-13, thm-13, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0090","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57966795f6912d00b081a651739915c84fac6d031a7a664094614f6224b33085","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197090,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":false,\"environment_id\":\"env-14\",\"kernel_checked\":true,\"statement_id\":\"thm-14\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000e\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000072\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-14\",\"statement_id\":\"thm-14\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000e\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000072\",\"kernel_checked\":true,\"axioms_recorded\":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-14, thm-14, Yes, Yes, Yes, No, Yes, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-14, thm-14, Yes, Yes, Yes, No, Yes, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0091","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4ca676e89f3b4ad5ee7f4312448c6bfa22b6170f9e35096dd0016d6564c8baa","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197091,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-15\",\"kernel_checked\":false,\"statement_id\":\"\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000f\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-15\",\"statement_id\":\"\",\"term_sha256\":\"000000000000000000000000000000000000000000000000000000000000000f\",\"type_sha256\":\"bad\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-15, missing, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-15, missing, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0092","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1abbc95e44570a05ef9b1e70491fa84ebb2854b5f08d088397807aacf2a4186","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197092,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-16\",\"kernel_checked\":true,\"statement_id\":\"thm-16\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000074\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-16\",\"statement_id\":\"thm-16\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000074\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-16, thm-16, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-16, thm-16, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0093","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c95ee921be31efbd5ef4d0a1ed14aa7460accbf9d4d6bc80a9a6b168e769c1","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197093,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-17\",\"kernel_checked\":false,\"statement_id\":\"thm-17\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000011\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000075\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-17\",\"statement_id\":\"thm-17\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000011\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000075\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-17, thm-17, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-17, thm-17, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0094","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e029619a57d1c7de23674a963548d298556dec46569e300be9f0c7c612aae02","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197094,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"\",\"kernel_checked\":true,\"statement_id\":\"thm-18\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000012\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"\",\"statement_id\":\"thm-18\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000012\",\"type_sha256\":\"bad\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(missing, thm-18, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-18, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0095","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90ba444b40b9be0db51b02c82443e3d009ec8d93a94eba8f18e084939c64c7f8","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197095,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-19\",\"kernel_checked\":false,\"statement_id\":\"thm-19\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000013\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000077\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-19\",\"statement_id\":\"thm-19\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000013\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000077\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-19, thm-19, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-19, thm-19, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0096","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7e98b806009b57f310e6cb2bc25e8a84ab75efa042dfc930ab6d13c9e6a0b27","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197096,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-20\",\"kernel_checked\":true,\"statement_id\":\"\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000078\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-20\",\"statement_id\":\"\",\"term_sha256\":\"bad\",\"type_sha256\":\"0000000000000000000000000000000000000000000000000000000000000078\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-20, missing, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-20, missing, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0097","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39e14eb36ecac34fa3dd4bbb6b60e9731b3b846ad38e123a27eea57dec892d5","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197097,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":false,\"environment_id\":\"env-21\",\"kernel_checked\":false,\"statement_id\":\"thm-21\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000015\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-21\",\"statement_id\":\"thm-21\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000015\",\"type_sha256\":\"bad\",\"kernel_checked\":false,\"axioms_recorded\":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-21, thm-21, Yes, No, No, No, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-21, thm-21, Yes, No, No, No, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0098","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b24197b447c7811eae748341e4ed34bfd1a965cf1a866d0084c6ce0c3ccf184","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197098,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-22\",\"kernel_checked\":true,\"statement_id\":\"thm-22\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000016\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000007a\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-22\",\"statement_id\":\"thm-22\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000016\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000007a\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-22, thm-22, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-22, thm-22, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0099","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d949c9d24f89ca27d5fc0ec646c4c58d0f444f2543bae8b95f092636a0691d62","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197099,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"env-23\",\"kernel_checked\":false,\"statement_id\":\"thm-23\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000017\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000007b\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"env-23\",\"statement_id\":\"thm-23\",\"term_sha256\":\"0000000000000000000000000000000000000000000000000000000000000017\",\"type_sha256\":\"000000000000000000000000000000000000000000000000000000000000007b\",\"kernel_checked\":false,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(env-23, thm-23, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-23, thm-23, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r07-lean-proof-term-0100","batch_id":"batch-20261013-advance-20261019-r07-lean-proof-term-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf1af20609a249666d6eba69508cfdf9bba260322109c0834074dd3cf121c3c","family_id":"qf-ext19-proofterm-kernel-receipt","seed":202610197100,"semantic_template":"qf-ext19-proofterm-kernel-receipt/{\"axioms_recorded\":true,\"environment_id\":\"\",\"kernel_checked\":true,\"statement_id\":\"thm-24\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Kernel Metadata Qualification and Axiom Inventory Flag","stem":"Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={\"environment_id\":\"\",\"statement_id\":\"thm-24\",\"term_sha256\":\"bad\",\"type_sha256\":\"bad\",\"kernel_checked\":true,\"axioms_recorded\":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.","answer":{"kind":"expression","value":"(missing, thm-24, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)","explanation":"Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-24, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)."},"skills":["Synthetic Kernel Metadata Qualification and Axiom Inventory Flag"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}