{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261018-r07-lean-formal-statement-0001","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59ffa72786ab0a1050923dddc7ab8337351b862b2f9cddd9d1cb20da702b0ecb","family_id":"qf-ext18-formal-pipeline-state","seed":202610187001,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-0\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-0\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-0, [No, No, No, No], Yes, parse-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-0, [No, No, No, No], Yes, parse-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0002","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3c6034abc70148f9c538bac9d761ec98417549f5475093a5ab69bfc604722b7","family_id":"qf-ext18-formal-pipeline-state","seed":202610187002,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-1\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-1\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-1, [Yes, No, No, No], Yes, elaboration-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-1, [Yes, No, No, No], Yes, elaboration-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0003","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c5d4b685e4e147fe7786112f37a334ac9117e4848dbddbef59b53f75916553e","family_id":"qf-ext18-formal-pipeline-state","seed":202610187003,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-2\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-2\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-2, [Yes, Yes, No, No], Yes, type-unknown)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-2, [Yes, Yes, No, No], Yes, type-unknown)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0004","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99226e748d0a8cc8eeab38d5d68479cc8fdd01bfb300f78419284fdd173ae790","family_id":"qf-ext18-formal-pipeline-state","seed":202610187004,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-3\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-3\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-3, [Yes, Yes, Yes, No], Yes, not-Prop)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-3, [Yes, Yes, Yes, No], Yes, not-Prop)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0005","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb48b5edde11bcdec1f4729b77ac69d0bc027f0d0cfc972be09fa42d18eaac1c","family_id":"qf-ext18-formal-pipeline-state","seed":202610187005,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-4\",\"parsed\":true,\"type_is_prop\":true,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-4\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, true]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-4, [Yes, Yes, Yes, Yes], Yes, formal-statement)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-4, [Yes, Yes, Yes, Yes], Yes, formal-statement)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0006","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23c9410b712225fd4d7c1a1aa34b08ac7b8c64b1d2c6ff27ab1d909e5dd5bb5d","family_id":"qf-ext18-formal-pipeline-state","seed":202610187006,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-5\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-5\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-5, [No, Yes, No, No], No, inconsistent-receipt)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-5, [No, Yes, No, No], No, inconsistent-receipt)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0007","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92b741348596aad613a3be6c28cdb200dfed7abe1a355870f7ece970a5ac2ed","family_id":"qf-ext18-formal-pipeline-state","seed":202610187007,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-6\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-6\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-6, [No, No, No, No], Yes, parse-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-6, [No, No, No, No], Yes, parse-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0008","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8627448347ad9c84879d55ef156e49d1769f944cba2c1e1b8227615947aec8df","family_id":"qf-ext18-formal-pipeline-state","seed":202610187008,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-7\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-7\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-7, [Yes, No, No, No], Yes, elaboration-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-7, [Yes, No, No, No], Yes, elaboration-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0009","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92266a470b6085ebe3b6dbfd44c93d76995a20a31692db01bdbd0435c402cc2a","family_id":"qf-ext18-formal-pipeline-state","seed":202610187009,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-8\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-8\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-8, [Yes, Yes, No, No], Yes, type-unknown)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-8, [Yes, Yes, No, No], Yes, type-unknown)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0010","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed92e8cb3b4997d4b329052888b81b071c516778ce8b7b3fe4983e583494db97","family_id":"qf-ext18-formal-pipeline-state","seed":202610187010,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-9\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-9\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-9, [Yes, Yes, Yes, No], Yes, not-Prop)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-9, [Yes, Yes, Yes, No], Yes, not-Prop)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0011","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9da8b3896c2f586926ecfe082288c26c0712f40c7eb14ac680f33ac005be662","family_id":"qf-ext18-formal-pipeline-state","seed":202610187011,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-10\",\"parsed\":true,\"type_is_prop\":true,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-10\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, true]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-10, [Yes, Yes, Yes, Yes], Yes, formal-statement)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-10, [Yes, Yes, Yes, Yes], Yes, formal-statement)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0012","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"265746129bfb0f9e7a308d2eb90411a7c433267c6049fd9124520dee9c56a2bb","family_id":"qf-ext18-formal-pipeline-state","seed":202610187012,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-11\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-11\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-11, [No, Yes, No, No], No, inconsistent-receipt)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-11, [No, Yes, No, No], No, inconsistent-receipt)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0013","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecf3d35030a654e110b5e8456f5282f8999d80e3a8c84c2938ac7a8533f45ca3","family_id":"qf-ext18-formal-pipeline-state","seed":202610187013,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-12\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-12\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-12, [No, No, No, No], Yes, parse-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-12, [No, No, No, No], Yes, parse-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0014","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7147971ccf8c3b4d831f7b6c4c89d3af74750fd186f8dffa8fd3c526f570d051","family_id":"qf-ext18-formal-pipeline-state","seed":202610187014,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-13\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-13\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-13, [Yes, No, No, No], Yes, elaboration-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-13, [Yes, No, No, No], Yes, elaboration-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0015","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf456eeb4d724fdce697a407aff9e3566fcc4dac7ac27e42875762866e0243af","family_id":"qf-ext18-formal-pipeline-state","seed":202610187015,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-14\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-14\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-14, [Yes, Yes, No, No], Yes, type-unknown)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-14, [Yes, Yes, No, No], Yes, type-unknown)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0016","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e9a52c794dc0a912034c36511bea360afe2f44e70fb13b3b5e086c78fd4e3c","family_id":"qf-ext18-formal-pipeline-state","seed":202610187016,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-15\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-15\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-15, [Yes, Yes, Yes, No], Yes, not-Prop)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-15, [Yes, Yes, Yes, No], Yes, not-Prop)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0017","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"472b362b8725c7748d2c2d33ce6efd4e00073ec20feaf2f61d3ea1aa160ce0bb","family_id":"qf-ext18-formal-pipeline-state","seed":202610187017,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-16\",\"parsed\":true,\"type_is_prop\":true,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-16\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, true]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-16, [Yes, Yes, Yes, Yes], Yes, formal-statement)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-16, [Yes, Yes, Yes, Yes], Yes, formal-statement)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0018","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"276ada520fa6fa137c62ad9de8f02b75a3c3f4e0eb70371b9290822cba911546","family_id":"qf-ext18-formal-pipeline-state","seed":202610187018,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-17\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-17\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-17, [No, Yes, No, No], No, inconsistent-receipt)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-17, [No, Yes, No, No], No, inconsistent-receipt)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0019","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"154abc0de57f9d9918680e76c71e8173e15e027dac9087db71d9b697dce57d6d","family_id":"qf-ext18-formal-pipeline-state","seed":202610187019,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-18\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-18\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-18, [No, No, No, No], Yes, parse-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-18, [No, No, No, No], Yes, parse-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0020","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"566d403b29668a85a06d5d0ddcadeb5c55a3ab2b058da10152fbb8553785bec9","family_id":"qf-ext18-formal-pipeline-state","seed":202610187020,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-19\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-19\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-19, [Yes, No, No, No], Yes, elaboration-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-19, [Yes, No, No, No], Yes, elaboration-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0021","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9908d978a47debb19fd524c40f7b9b200575b9864a39e3559919d97f8b419efc","family_id":"qf-ext18-formal-pipeline-state","seed":202610187021,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-20\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-20\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-20, [Yes, Yes, No, No], Yes, type-unknown)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-20, [Yes, Yes, No, No], Yes, type-unknown)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0022","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62c98a7295a2c88aab7e122d9fb03c86b7e70877482066d4c0ff99324742c51b","family_id":"qf-ext18-formal-pipeline-state","seed":202610187022,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-21\",\"parsed\":true,\"type_is_prop\":false,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-21\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-21, [Yes, Yes, Yes, No], Yes, not-Prop)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-21, [Yes, Yes, Yes, No], Yes, not-Prop)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0023","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf3e636c7b31ada1a71e7950af88e2713cc1d50425d99ad4e1d64e111e32f3c","family_id":"qf-ext18-formal-pipeline-state","seed":202610187023,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-22\",\"parsed\":true,\"type_is_prop\":true,\"type_known\":true}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-22\"; flags [parsed,elaborated,type_known,type_is_prop]=[true, true, true, true]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-22, [Yes, Yes, Yes, Yes], Yes, formal-statement)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-22, [Yes, Yes, Yes, Yes], Yes, formal-statement)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0024","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3618ed970b810cfaf24e9718eb3cf21f43eb5c232a76779c38065447e5d6da","family_id":"qf-ext18-formal-pipeline-state","seed":202610187024,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":true,\"environment_id\":\"env-23\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-23\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, true, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-23, [No, Yes, No, No], No, inconsistent-receipt)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-23, [No, Yes, No, No], No, inconsistent-receipt)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0025","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"288796ca67912ea0c90520876975fb920d8506151cb20f5b29493693f05adef5","family_id":"qf-ext18-formal-pipeline-state","seed":202610187025,"semantic_template":"qf-ext18-formal-pipeline-state/{\"elaborated\":false,\"environment_id\":\"env-24\",\"parsed\":false,\"type_is_prop\":false,\"type_known\":false}"},"fields":{"title":"Synthetic Formal-Statement Pipeline Stage and Coherence","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-24\"; flags [parsed,elaborated,type_known,type_is_prop]=[false, false, false, false]. Give (environment ID, flags, coherence?, stage label). Later-stage success requires earlier-stage success.","answer":{"kind":"expression","value":"(env-24, [No, No, No, No], Yes, parse-failed)","explanation":"Classify the supplied synthetic flags; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-24, [No, No, No, No], Yes, parse-failed)."},"skills":["Synthetic Formal-Statement Pipeline Stage and Coherence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0026","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd4dc78d3396c49dab94067da951c0d1e3a9a13233796277c4720b3a5786e413","family_id":"qf-ext18-formal-context-inventory","seed":202610187026,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-0\",\"required_names\":[\"Nat\",\"x0\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-0\", required_names=[\"Nat\",\"x0\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-0, [Nat, x0], [h, Nat], [x0], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-0, [Nat, x0], [h, Nat], [x0], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0027","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c60fbafecba5fb9be33621f5b20552e7787a55adb67b368c4f71e61967426ec7","family_id":"qf-ext18-formal-context-inventory","seed":202610187027,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x1\"],\"environment_id\":\"env-1\",\"required_names\":[\"Nat\",\"x1\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-1\", required_names=[\"Nat\",\"x1\"], available_names=[\"Nat\",\"x1\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-1, [Nat, x1], [h, Nat, x1], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-1, [Nat, x1], [h, Nat, x1], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0028","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60444cb7bf43b348b2a0967929afe48a6c468cfce748772d98506ffb2d3af8e4","family_id":"qf-ext18-formal-context-inventory","seed":202610187028,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-2\",\"required_names\":[\"Nat\",\"x2\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-2\", required_names=[\"Nat\",\"x2\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-2, [Nat, x2], [h, Nat], [x2], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-2, [Nat, x2], [h, Nat], [x2], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0029","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1624ebc7f2b6b7769f05d09171a97acccfe451d1adc5dbb4972b37a3ac3fcf64","family_id":"qf-ext18-formal-context-inventory","seed":202610187029,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x3\"],\"environment_id\":\"env-3\",\"required_names\":[\"Nat\",\"x3\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-3\", required_names=[\"Nat\",\"x3\"], available_names=[\"Nat\",\"x3\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-3, [Nat, x3], [h, Nat, x3], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-3, [Nat, x3], [h, Nat, x3], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0030","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb91142041664e120f62e67637d6ccbff58118d3aaaa3fe79b51442e5165ff25","family_id":"qf-ext18-formal-context-inventory","seed":202610187030,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-4\",\"required_names\":[\"Nat\",\"x4\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-4\", required_names=[\"Nat\",\"x4\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-4, [Nat, x4], [h, Nat], [x4], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-4, [Nat, x4], [h, Nat], [x4], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0031","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8adf808704e70d36abad58ce34d71100d7bc085ffdcf149777b640b42420500","family_id":"qf-ext18-formal-context-inventory","seed":202610187031,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x5\"],\"environment_id\":\"env-5\",\"required_names\":[\"Nat\",\"x5\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-5\", required_names=[\"Nat\",\"x5\"], available_names=[\"Nat\",\"x5\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-5, [Nat, x5], [h, Nat, x5], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-5, [Nat, x5], [h, Nat, x5], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0032","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e660b1539df762a43a564322ad8ffc74d70fd305af87643d977ba2fd8545582","family_id":"qf-ext18-formal-context-inventory","seed":202610187032,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-6\",\"required_names\":[\"Nat\",\"x6\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-6\", required_names=[\"Nat\",\"x6\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-6, [Nat, x6], [h, Nat], [x6], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-6, [Nat, x6], [h, Nat], [x6], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0033","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e2729fcf00b8f4a5e2efbce80f8d97b96ac9087bfa6e9bfa83ed5dd1512ec0b","family_id":"qf-ext18-formal-context-inventory","seed":202610187033,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x7\"],\"environment_id\":\"env-7\",\"required_names\":[\"Nat\",\"x7\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-7\", required_names=[\"Nat\",\"x7\"], available_names=[\"Nat\",\"x7\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-7, [Nat, x7], [h, Nat, x7], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-7, [Nat, x7], [h, Nat, x7], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0034","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8945495d45c1d9c8783b57d7e7e65ecf83f5d6efb3287ef938fc820b35f85fc7","family_id":"qf-ext18-formal-context-inventory","seed":202610187034,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-8\",\"required_names\":[\"Nat\",\"x8\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-8\", required_names=[\"Nat\",\"x8\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-8, [Nat, x8], [h, Nat], [x8], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-8, [Nat, x8], [h, Nat], [x8], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0035","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24cae644cad0d4fbe1758eb3c46b97c96e4c465534a74a776b359a7d37ac8b94","family_id":"qf-ext18-formal-context-inventory","seed":202610187035,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x9\"],\"environment_id\":\"env-9\",\"required_names\":[\"Nat\",\"x9\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-9\", required_names=[\"Nat\",\"x9\"], available_names=[\"Nat\",\"x9\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-9, [Nat, x9], [h, Nat, x9], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-9, [Nat, x9], [h, Nat, x9], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0036","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dad51dd87505cce4115a465468fd9381eaa1c30587344550553de3c72d7485ba","family_id":"qf-ext18-formal-context-inventory","seed":202610187036,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-10\",\"required_names\":[\"Nat\",\"x10\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-10\", required_names=[\"Nat\",\"x10\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-10, [Nat, x10], [h, Nat], [x10], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-10, [Nat, x10], [h, Nat], [x10], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0037","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"797ac0cdf0e3e133c99a4aaf700e73a2c4b1418b7ff83a3a6445a77fee546cc3","family_id":"qf-ext18-formal-context-inventory","seed":202610187037,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x11\"],\"environment_id\":\"env-11\",\"required_names\":[\"Nat\",\"x11\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-11\", required_names=[\"Nat\",\"x11\"], available_names=[\"Nat\",\"x11\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-11, [Nat, x11], [h, Nat, x11], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-11, [Nat, x11], [h, Nat, x11], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0038","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7658ce745483f1f88e4fa9837489674d3d35d52c5516d1eb17cc4508fe2c97f5","family_id":"qf-ext18-formal-context-inventory","seed":202610187038,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-12\",\"required_names\":[\"Nat\",\"x12\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-12\", required_names=[\"Nat\",\"x12\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-12, [Nat, x12], [h, Nat], [x12], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-12, [Nat, x12], [h, Nat], [x12], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0039","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cd4aac8df85ef17d048201ea26acfc8fbf9e5b894faff766e0c09f495f206aa","family_id":"qf-ext18-formal-context-inventory","seed":202610187039,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x13\"],\"environment_id\":\"env-13\",\"required_names\":[\"Nat\",\"x13\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-13\", required_names=[\"Nat\",\"x13\"], available_names=[\"Nat\",\"x13\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-13, [Nat, x13], [h, Nat, x13], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-13, [Nat, x13], [h, Nat, x13], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0040","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c0877eed83debc759e016cb95b8c5b6f82956220e87d80a0aa70b41d25b2dcc","family_id":"qf-ext18-formal-context-inventory","seed":202610187040,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-14\",\"required_names\":[\"Nat\",\"x14\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-14\", required_names=[\"Nat\",\"x14\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-14, [Nat, x14], [h, Nat], [x14], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-14, [Nat, x14], [h, Nat], [x14], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0041","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18bb61281971834dd620cfbcbe15ec3f69a86e67157511f719a906045108ec46","family_id":"qf-ext18-formal-context-inventory","seed":202610187041,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x15\"],\"environment_id\":\"env-15\",\"required_names\":[\"Nat\",\"x15\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-15\", required_names=[\"Nat\",\"x15\"], available_names=[\"Nat\",\"x15\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-15, [Nat, x15], [h, Nat, x15], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-15, [Nat, x15], [h, Nat, x15], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0042","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8e5e16add6c3ffe0f38580c7cc030339ae04e3837c1407467e3897a210ec840","family_id":"qf-ext18-formal-context-inventory","seed":202610187042,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-16\",\"required_names\":[\"Nat\",\"x16\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-16\", required_names=[\"Nat\",\"x16\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-16, [Nat, x16], [h, Nat], [x16], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-16, [Nat, x16], [h, Nat], [x16], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0043","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b6d698ba4f30cc306a5ff22f1854c47b47521da78b36a7c55c709426e3b0959","family_id":"qf-ext18-formal-context-inventory","seed":202610187043,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x17\"],\"environment_id\":\"env-17\",\"required_names\":[\"Nat\",\"x17\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-17\", required_names=[\"Nat\",\"x17\"], available_names=[\"Nat\",\"x17\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-17, [Nat, x17], [h, Nat, x17], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-17, [Nat, x17], [h, Nat, x17], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0044","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"434e8382ff920c3f03ef51887f1e01bef92011b1080044e1538efd8122e30fb0","family_id":"qf-ext18-formal-context-inventory","seed":202610187044,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-18\",\"required_names\":[\"Nat\",\"x18\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-18\", required_names=[\"Nat\",\"x18\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-18, [Nat, x18], [h, Nat], [x18], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-18, [Nat, x18], [h, Nat], [x18], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0045","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbeeb258c6bc9f258c997eb8bacf4ab843bc7a87cb6ee6b2e0f1adce8cc8181a","family_id":"qf-ext18-formal-context-inventory","seed":202610187045,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x19\"],\"environment_id\":\"env-19\",\"required_names\":[\"Nat\",\"x19\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-19\", required_names=[\"Nat\",\"x19\"], available_names=[\"Nat\",\"x19\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-19, [Nat, x19], [h, Nat, x19], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-19, [Nat, x19], [h, Nat, x19], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0046","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79babe391a157f485877d0303690b1ae08e38b9f93a4f0ae47f34a4ae20dace3","family_id":"qf-ext18-formal-context-inventory","seed":202610187046,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-20\",\"required_names\":[\"Nat\",\"x20\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-20\", required_names=[\"Nat\",\"x20\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-20, [Nat, x20], [h, Nat], [x20], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-20, [Nat, x20], [h, Nat], [x20], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0047","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"193de4675ed8ca0ebf2c84105de4bf8c806df00c426c43a06a7e24704766fa0e","family_id":"qf-ext18-formal-context-inventory","seed":202610187047,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x21\"],\"environment_id\":\"env-21\",\"required_names\":[\"Nat\",\"x21\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-21\", required_names=[\"Nat\",\"x21\"], available_names=[\"Nat\",\"x21\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-21, [Nat, x21], [h, Nat, x21], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-21, [Nat, x21], [h, Nat, x21], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0048","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a9e2db39322b3285527ad8273b264823ff3191ecca662c60bbd9df28cdefce3","family_id":"qf-ext18-formal-context-inventory","seed":202610187048,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-22\",\"required_names\":[\"Nat\",\"x22\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-22\", required_names=[\"Nat\",\"x22\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-22, [Nat, x22], [h, Nat], [x22], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-22, [Nat, x22], [h, Nat], [x22], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0049","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf17b62deb9026bda55d077b2c6c0ee946a15f9901a984592e5d2e7a3b8219d8","family_id":"qf-ext18-formal-context-inventory","seed":202610187049,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\",\"x23\"],\"environment_id\":\"env-23\",\"required_names\":[\"Nat\",\"x23\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-23\", required_names=[\"Nat\",\"x23\"], available_names=[\"Nat\",\"x23\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-23, [Nat, x23], [h, Nat, x23], [], Yes)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-23, [Nat, x23], [h, Nat, x23], [], Yes)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0050","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd2573749bde56ab20b3621104d08addcb3cf129e4bbd744f4ef99bda8cd0e1","family_id":"qf-ext18-formal-context-inventory","seed":202610187050,"semantic_template":"qf-ext18-formal-context-inventory/{\"available_names\":[\"h\",\"Nat\"],\"environment_id\":\"env-24\",\"required_names\":[\"Nat\",\"x24\"]}"},"fields":{"title":"Synthetic Required and Available Context Names","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-24\", required_names=[\"Nat\",\"x24\"], available_names=[\"Nat\",\"h\"]. Give (environment ID, sorted distinct required names, sorted distinct available names, missing required names, complete?). Use the specified English-locale name order and retain opaque names exactly.","answer":{"kind":"expression","value":"(env-24, [Nat, x24], [h, Nat], [x24], No)","explanation":"Compare explicit name inventories only; this is not an actual Lean name-resolution or elaboration run. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-24, [Nat, x24], [h, Nat], [x24], No)."},"skills":["Synthetic Required and Available Context Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0051","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8be8ecacb23081094ef5a31e280d99475dc4142ca6f7486f7bf02c0ee632571","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187051,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-0\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-0\", statement_elaborated_prop=false, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-0, No, No, No, Yes, no-formal-statement)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-0, No, No, No, Yes, no-formal-statement)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0052","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e3f8412f72591b0f733dffff64f78e4ee1264c5ea72f1203d29a707d78ef494","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187052,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-1\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-1\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-1, Yes, No, No, Yes, statement-only)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-1, Yes, No, No, Yes, statement-only)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0053","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90970da6fab1ee461d6c7bb8dfb4acd062f735cc90c5099287152b6925b08f06","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187053,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-2\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-2\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-2, Yes, Yes, No, Yes, unchecked-proof-candidate)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-2, Yes, Yes, No, Yes, unchecked-proof-candidate)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0054","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5158864b67492ebb832121ae3d5ea1c6428c04bf4b13c45ac37137430bcdc944","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187054,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-3\",\"kernel_checked\":true,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-3\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-3, Yes, Yes, Yes, Yes, has-kernel receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-3, Yes, Yes, Yes, Yes, has-kernel receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0055","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25ae0832545564ebccb9a1d46a593214e5592182f8133b8635093a966a4fcf26","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187055,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-4\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-4\", statement_elaborated_prop=false, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-4, No, Yes, No, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-4, No, Yes, No, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0056","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb152523432bf60bdcb457ac3f9ae6725de8babc8129578406ce891259260355","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187056,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-5\",\"kernel_checked\":true,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-5\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-5, Yes, No, Yes, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-5, Yes, No, Yes, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0057","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0516c3e3907bf83c24ec0b9004079f88dd85dd9325a23184ebd90f4c0da76c8","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187057,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-6\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-6\", statement_elaborated_prop=false, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-6, No, No, No, Yes, no-formal-statement)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-6, No, No, No, Yes, no-formal-statement)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0058","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6022211a150d544d2e05b06e9a8b7e94bff3890f8db55eb956ea08dfa00fad2c","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187058,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-7\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-7\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-7, Yes, No, No, Yes, statement-only)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-7, Yes, No, No, Yes, statement-only)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0059","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa57be689295f8bcfdaa82998ad31355d737c297f8dcc5c99d5b7953a484629d","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187059,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-8\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-8\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-8, Yes, Yes, No, Yes, unchecked-proof-candidate)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-8, Yes, Yes, No, Yes, unchecked-proof-candidate)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0060","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bde8857d74cb9eb19c85e6832405e2bbc47e1504cc0f189765ee096894b3547f","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187060,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-9\",\"kernel_checked\":true,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-9\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-9, Yes, Yes, Yes, Yes, has-kernel receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-9, Yes, Yes, Yes, Yes, has-kernel receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0061","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb64caea2d81b5c5d13009ec29e61f3d86c3fbfa4b1465e0cf3ec0c4d5c562bb","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187061,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-10\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-10\", statement_elaborated_prop=false, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-10, No, Yes, No, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-10, No, Yes, No, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0062","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"015268cfc752e89c51e5de48aa35e8efa8bd9101993161d931de90bbe16d7cce","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187062,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-11\",\"kernel_checked\":true,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-11\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-11, Yes, No, Yes, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-11, Yes, No, Yes, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0063","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67704f3f546db0c85229ba3ded9a1793332c5e3eabb75ed4f01eb582a77ad6d6","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187063,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-12\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-12\", statement_elaborated_prop=false, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-12, No, No, No, Yes, no-formal-statement)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-12, No, No, No, Yes, no-formal-statement)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0064","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9dc32b580c4bfacc85d7ad68c7be3c18be7c6a11021203643cf2d3c5884bfdd","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187064,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-13\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-13\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-13, Yes, No, No, Yes, statement-only)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-13, Yes, No, No, Yes, statement-only)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0065","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4601ed4dc1e579683066d65554266edc33c76f72e7c38874394e6c1a98f8483","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187065,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-14\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-14\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-14, Yes, Yes, No, Yes, unchecked-proof-candidate)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-14, Yes, Yes, No, Yes, unchecked-proof-candidate)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0066","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d5d4235f5e5b9df91388a6e3ca9d090525dd6699eecbd0a3f898ce9b004ddec","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187066,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-15\",\"kernel_checked\":true,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-15\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-15, Yes, Yes, Yes, Yes, has-kernel receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-15, Yes, Yes, Yes, Yes, has-kernel receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0067","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59aad3e7b61eec334be6bf326120d86e5920bd88152471bade07420421089981","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187067,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-16\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-16\", statement_elaborated_prop=false, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-16, No, Yes, No, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-16, No, Yes, No, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0068","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9addc0ea984dc72267570bd3e75badec6b9a3a559c4e3bc3401c2507729284","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187068,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-17\",\"kernel_checked\":true,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-17\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-17, Yes, No, Yes, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-17, Yes, No, Yes, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0069","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f10fa31d9ac00074c0fb5f0cb0bd3e8ce6747b2789f2e5409e3222b3f8cf7c1a","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187069,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-18\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-18\", statement_elaborated_prop=false, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-18, No, No, No, Yes, no-formal-statement)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-18, No, No, No, Yes, no-formal-statement)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0070","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3df774c0c17aadf783b124f56d9f36f5151b719a4b4cb80ef5ba5548bc4c5dce","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187070,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-19\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-19\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-19, Yes, No, No, Yes, statement-only)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-19, Yes, No, No, Yes, statement-only)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0071","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f12ac4c9f7b85047f4878e63a2eb403190430310650258011b8035cf26e6c01","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187071,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-20\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-20\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-20, Yes, Yes, No, Yes, unchecked-proof-candidate)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-20, Yes, Yes, No, Yes, unchecked-proof-candidate)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0072","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5356583fb3697990fb127a1cc0c82329fa7c8fcc44514f3234220eeefbeda90","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187072,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-21\",\"kernel_checked\":true,\"proof_term_present\":true,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-21\", statement_elaborated_prop=true, proof_term_present=true, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-21, Yes, Yes, Yes, Yes, has-kernel receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-21, Yes, Yes, Yes, Yes, has-kernel receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0073","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e148cbc19afb820794bff41ecceba662096064d3dc14337167f73d2b07cec40","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187073,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-22\",\"kernel_checked\":false,\"proof_term_present\":true,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-22\", statement_elaborated_prop=false, proof_term_present=true, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-22, No, Yes, No, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-22, No, Yes, No, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0074","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7804caefe37e7c8642bff5d736d0caa623293425005fbcf675d9e532bff2442","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187074,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-23\",\"kernel_checked\":true,\"proof_term_present\":false,\"statement_elaborated_prop\":true}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-23\", statement_elaborated_prop=true, proof_term_present=false, kernel_checked=true. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-23, Yes, No, Yes, No, inconsistent-receipt)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-23, Yes, No, Yes, No, inconsistent-receipt)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0075","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f93f5bcaa885381ec564020e6b1c3c148485c736f67ffff0ca69bed1bba8f65","family_id":"qf-ext18-formal-statement-proof-separation","seed":202610187075,"semantic_template":"qf-ext18-formal-statement-proof-separation/{\"environment_id\":\"env-24\",\"kernel_checked\":false,\"proof_term_present\":false,\"statement_elaborated_prop\":false}"},"fields":{"title":"Synthetic Statement, Proof-Term, and Kernel Flags","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. environment_id=\"env-24\", statement_elaborated_prop=false, proof_term_present=false, kernel_checked=false. Give (environment ID, those three flags, coherence?, state label). A proof flag requires a statement, and a kernel flag requires a proof term.","answer":{"kind":"expression","value":"(env-24, No, No, No, Yes, no-formal-statement)","explanation":"Distinguish statement-only, unchecked proof candidate and a supplied kernel-receipt flag; this is not an actual Lean execution. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-24, No, No, No, Yes, no-formal-statement)."},"skills":["Synthetic Statement, Proof-Term, and Kernel Flags"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0076","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b46153badff3398a9c0f56a11f2a2f64a660cd0a6c79436d6ed4fe20c955735","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187076,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"\",\"core_type\":\"Type\",\"elaboration_status\":\"failed\",\"environment_id\":\"\",\"imports\":[],\"imports_declared\":false,\"lean_version\":\"\",\"source_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"\",\"imports\":[],\"imports_declared\":false,\"environment_id\":\"\",\"context_id\":\"\",\"source_sha256\":\"bad\",\"core_type\":\"Type\",\"elaboration_status\":\"failed\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(missing, [], No, missing, missing, No, Type, failed, [lean_version, imports_declaration, environment_id, context_id, source_sha256, core_type_Prop, elaboration_success], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, [], No, missing, missing, No, Type, failed, [lean_version, imports_declaration, environment_id, context_id, source_sha256, core_type_Prop, elaboration_success], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0077","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a53703e7d265c55bfbd89c88fdc47572e7b94caf74ed009ad5164e0ffd76872a","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187077,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-1\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-1\",\"imports\":[\"Mathlib\",\"Module1\"],\"imports_declared\":true,\"lean_version\":\"4.21.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000001\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.21.0\",\"imports\":[\"Mathlib\",\"Module1\"],\"imports_declared\":true,\"environment_id\":\"env-1\",\"context_id\":\"ctx-1\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000001\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.21.0, [Mathlib, Module1], Yes, env-1, ctx-1, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.21.0, [Mathlib, Module1], Yes, env-1, ctx-1, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0078","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5bd13981f8f46a75433fb3ac3057571190431738bd7724dc28296a12f8a06d","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187078,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-2\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-2\",\"imports\":[\"Mathlib\",\"Module2\"],\"imports_declared\":true,\"lean_version\":\"4.22.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000002\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.22.0\",\"imports\":[\"Mathlib\",\"Module2\"],\"imports_declared\":true,\"environment_id\":\"env-2\",\"context_id\":\"ctx-2\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000002\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.22.0, [Mathlib, Module2], Yes, env-2, ctx-2, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.22.0, [Mathlib, Module2], Yes, env-2, ctx-2, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0079","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5671b09c1163c58507ae031a4e0fb3e14432db6bd3a0d1efff19028ae68bddc","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187079,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-3\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-3\",\"imports\":[\"Mathlib\",\"Module3\"],\"imports_declared\":true,\"lean_version\":\"4.23.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000003\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.23.0\",\"imports\":[\"Mathlib\",\"Module3\"],\"imports_declared\":true,\"environment_id\":\"env-3\",\"context_id\":\"ctx-3\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000003\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.23.0, [Mathlib, Module3], Yes, env-3, ctx-3, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.23.0, [Mathlib, Module3], Yes, env-3, ctx-3, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0080","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a55b4edd4a4192b7e5cb765dce58b0da81606a1f24012fb0e341d024887a09ac","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187080,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-4\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-4\",\"imports\":[],\"imports_declared\":true,\"lean_version\":\"4.24.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000004\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.24.0\",\"imports\":[],\"imports_declared\":true,\"environment_id\":\"env-4\",\"context_id\":\"ctx-4\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000004\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.24.0, [], Yes, env-4, ctx-4, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.24.0, [], Yes, env-4, ctx-4, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0081","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cd8cee7a10ed50663bc614ee3f881dd718b1440a7bd2b137869e0fec84281fd","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187081,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-5\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-5\",\"imports\":[\"Mathlib\",\"Module5\"],\"imports_declared\":true,\"lean_version\":\"\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000005\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"\",\"imports\":[\"Mathlib\",\"Module5\"],\"imports_declared\":true,\"environment_id\":\"env-5\",\"context_id\":\"ctx-5\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000005\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(missing, [Mathlib, Module5], Yes, env-5, ctx-5, Yes, Prop, success, [lean_version], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, [Mathlib, Module5], Yes, env-5, ctx-5, Yes, Prop, success, [lean_version], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0082","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfc081a6dac8d138ccead6739dab0fe3717c9ac1d528b27ba836447e43238375","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187082,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-6\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-6\",\"imports\":[\"Mathlib\",\"Module6\"],\"imports_declared\":false,\"lean_version\":\"4.26.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000006\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.26.0\",\"imports\":[\"Mathlib\",\"Module6\"],\"imports_declared\":false,\"environment_id\":\"env-6\",\"context_id\":\"ctx-6\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000006\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.26.0, [Mathlib, Module6], No, env-6, ctx-6, Yes, Prop, success, [imports_declaration], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.26.0, [Mathlib, Module6], No, env-6, ctx-6, Yes, Prop, success, [imports_declaration], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0083","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffb64eb91a3e80a318ac4b0342ea24a40c1d9225e692b92a70464e85e339a326","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187083,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-7\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"\",\"imports\":[\"Mathlib\",\"Module7\"],\"imports_declared\":true,\"lean_version\":\"4.27.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000007\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.27.0\",\"imports\":[\"Mathlib\",\"Module7\"],\"imports_declared\":true,\"environment_id\":\"\",\"context_id\":\"ctx-7\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000007\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.27.0, [Mathlib, Module7], Yes, missing, ctx-7, Yes, Prop, success, [environment_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.27.0, [Mathlib, Module7], Yes, missing, ctx-7, Yes, Prop, success, [environment_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0084","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46638cace891ecf25b40267ba5d7e99923eb7172e3798a8a8b0a13158d805102","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187084,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-8\",\"imports\":[],\"imports_declared\":true,\"lean_version\":\"4.28.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000008\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.28.0\",\"imports\":[],\"imports_declared\":true,\"environment_id\":\"env-8\",\"context_id\":\"\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000008\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.28.0, [], Yes, env-8, missing, Yes, Prop, success, [context_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.28.0, [], Yes, env-8, missing, Yes, Prop, success, [context_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0085","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96449ebe57d43cedf372430cb428a3705a5f956cf14dc281bd42a1c309e2da98","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187085,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-9\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-9\",\"imports\":[\"Mathlib\",\"Module9\"],\"imports_declared\":true,\"lean_version\":\"4.29.0\",\"source_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.29.0\",\"imports\":[\"Mathlib\",\"Module9\"],\"imports_declared\":true,\"environment_id\":\"env-9\",\"context_id\":\"ctx-9\",\"source_sha256\":\"bad\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.29.0, [Mathlib, Module9], Yes, env-9, ctx-9, No, Prop, success, [source_sha256], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.29.0, [Mathlib, Module9], Yes, env-9, ctx-9, No, Prop, success, [source_sha256], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0086","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b354fd3f788c3301f3af80e75b30aa55687adb4b83bbb73b927425f00fc669e2","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187086,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-10\",\"core_type\":\"Type\",\"elaboration_status\":\"success\",\"environment_id\":\"env-10\",\"imports\":[\"Mathlib\",\"Module10\"],\"imports_declared\":true,\"lean_version\":\"\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000a\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"\",\"imports\":[\"Mathlib\",\"Module10\"],\"imports_declared\":true,\"environment_id\":\"env-10\",\"context_id\":\"ctx-10\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000a\",\"core_type\":\"Type\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(missing, [Mathlib, Module10], Yes, env-10, ctx-10, Yes, Type, success, [lean_version, core_type_Prop], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, [Mathlib, Module10], Yes, env-10, ctx-10, Yes, Type, success, [lean_version, core_type_Prop], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0087","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74207d3da3fc6ac87ec0896fd5fdca439ff9cf01e68718830d311e0b8565dc46","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187087,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-11\",\"core_type\":\"Prop\",\"elaboration_status\":\"failed\",\"environment_id\":\"env-11\",\"imports\":[\"Mathlib\",\"Module11\"],\"imports_declared\":true,\"lean_version\":\"4.31.0\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000b\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.31.0\",\"imports\":[\"Mathlib\",\"Module11\"],\"imports_declared\":true,\"environment_id\":\"env-11\",\"context_id\":\"ctx-11\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000b\",\"core_type\":\"Prop\",\"elaboration_status\":\"failed\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.31.0, [Mathlib, Module11], Yes, env-11, ctx-11, Yes, Prop, failed, [elaboration_success], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.31.0, [Mathlib, Module11], Yes, env-11, ctx-11, Yes, Prop, failed, [elaboration_success], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0088","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c353822f9be3a1bdbdd1e199937bca874c1bb58193e5737e8280cc2fb16ca67","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187088,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-12\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-12\",\"imports\":[],\"imports_declared\":false,\"lean_version\":\"4.32.0\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000c\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.32.0\",\"imports\":[],\"imports_declared\":false,\"environment_id\":\"env-12\",\"context_id\":\"ctx-12\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000c\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.32.0, [], No, env-12, ctx-12, Yes, Prop, success, [imports_declaration], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.32.0, [], No, env-12, ctx-12, Yes, Prop, success, [imports_declaration], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0089","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7673046c43b7c35c71395cc5ab09563391e4dabc4b98c5a0cd84b0ca8643c5","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187089,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-13\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-13\",\"imports\":[\"Mathlib\",\"Module13\"],\"imports_declared\":true,\"lean_version\":\"4.33.0\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000d\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.33.0\",\"imports\":[\"Mathlib\",\"Module13\"],\"imports_declared\":true,\"environment_id\":\"env-13\",\"context_id\":\"ctx-13\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000d\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.33.0, [Mathlib, Module13], Yes, env-13, ctx-13, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.33.0, [Mathlib, Module13], Yes, env-13, ctx-13, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0090","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"364cae98a5b06dfb48a35f1073d72f495e85d623f52fd2339cbae04a977a3e30","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187090,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-14\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"\",\"imports\":[\"Mathlib\",\"Module14\"],\"imports_declared\":true,\"lean_version\":\"4.34.0\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000e\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.34.0\",\"imports\":[\"Mathlib\",\"Module14\"],\"imports_declared\":true,\"environment_id\":\"\",\"context_id\":\"ctx-14\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000e\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.34.0, [Mathlib, Module14], Yes, missing, ctx-14, Yes, Prop, success, [environment_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.34.0, [Mathlib, Module14], Yes, missing, ctx-14, Yes, Prop, success, [environment_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0091","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45351331cecc53f6a27b89d86b4bd6ca529c98951684b53c3ea5b804044c2d83","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187091,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-15\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-15\",\"imports\":[\"Mathlib\",\"Module15\"],\"imports_declared\":true,\"lean_version\":\"\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000f\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"\",\"imports\":[\"Mathlib\",\"Module15\"],\"imports_declared\":true,\"environment_id\":\"env-15\",\"context_id\":\"ctx-15\",\"source_sha256\":\"000000000000000000000000000000000000000000000000000000000000000f\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(missing, [Mathlib, Module15], Yes, env-15, ctx-15, Yes, Prop, success, [lean_version], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, [Mathlib, Module15], Yes, env-15, ctx-15, Yes, Prop, success, [lean_version], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0092","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39ca93b5ac137bd30e59c7850fb621b421ad2af7870bd1faa185ffd0600a8180","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187092,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-16\",\"imports\":[],\"imports_declared\":true,\"lean_version\":\"4.36.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000010\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.36.0\",\"imports\":[],\"imports_declared\":true,\"environment_id\":\"env-16\",\"context_id\":\"\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000010\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.36.0, [], Yes, env-16, missing, Yes, Prop, success, [context_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.36.0, [], Yes, env-16, missing, Yes, Prop, success, [context_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0093","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49c49983311ec43763e03d26cb43ee106c6f3c79d8d94241dfc67cc968ddb718","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187093,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-17\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-17\",\"imports\":[\"Mathlib\",\"Module17\"],\"imports_declared\":true,\"lean_version\":\"4.37.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000011\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.37.0\",\"imports\":[\"Mathlib\",\"Module17\"],\"imports_declared\":true,\"environment_id\":\"env-17\",\"context_id\":\"ctx-17\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000011\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.37.0, [Mathlib, Module17], Yes, env-17, ctx-17, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.37.0, [Mathlib, Module17], Yes, env-17, ctx-17, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0094","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1596f1ba5fff146c99290fe618597ff911e312fa1c72a1c3b4134a88b37596e0","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187094,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-18\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-18\",\"imports\":[\"Mathlib\",\"Module18\"],\"imports_declared\":false,\"lean_version\":\"4.38.0\",\"source_sha256\":\"bad\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.38.0\",\"imports\":[\"Mathlib\",\"Module18\"],\"imports_declared\":false,\"environment_id\":\"env-18\",\"context_id\":\"ctx-18\",\"source_sha256\":\"bad\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.38.0, [Mathlib, Module18], No, env-18, ctx-18, No, Prop, success, [imports_declaration, source_sha256], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.38.0, [Mathlib, Module18], No, env-18, ctx-18, No, Prop, success, [imports_declaration, source_sha256], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0095","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7984829fdd6a71e2c22129a93eeb69d2b39262d253f821d2b006ec6647f77a80","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187095,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-19\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-19\",\"imports\":[\"Mathlib\",\"Module19\"],\"imports_declared\":true,\"lean_version\":\"4.39.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000013\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.39.0\",\"imports\":[\"Mathlib\",\"Module19\"],\"imports_declared\":true,\"environment_id\":\"env-19\",\"context_id\":\"ctx-19\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000013\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.39.0, [Mathlib, Module19], Yes, env-19, ctx-19, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.39.0, [Mathlib, Module19], Yes, env-19, ctx-19, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0096","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d607056da265170cdb286c29dcd6beee9e6104b0ad072df1653830bd3498e3","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187096,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-20\",\"core_type\":\"Type\",\"elaboration_status\":\"success\",\"environment_id\":\"env-20\",\"imports\":[],\"imports_declared\":true,\"lean_version\":\"\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000014\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"\",\"imports\":[],\"imports_declared\":true,\"environment_id\":\"env-20\",\"context_id\":\"ctx-20\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000014\",\"core_type\":\"Type\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(missing, [], Yes, env-20, ctx-20, Yes, Type, success, [lean_version, core_type_Prop], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, [], Yes, env-20, ctx-20, Yes, Type, success, [lean_version, core_type_Prop], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0097","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8112580f34ddd63b6358db8fa962d5eb99b85f8b8667d955e4cda189fd268bdf","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187097,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-21\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"\",\"imports\":[\"Mathlib\",\"Module21\"],\"imports_declared\":true,\"lean_version\":\"4.41.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000015\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.41.0\",\"imports\":[\"Mathlib\",\"Module21\"],\"imports_declared\":true,\"environment_id\":\"\",\"context_id\":\"ctx-21\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000015\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.41.0, [Mathlib, Module21], Yes, missing, ctx-21, Yes, Prop, success, [environment_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.41.0, [Mathlib, Module21], Yes, missing, ctx-21, Yes, Prop, success, [environment_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0098","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0427c0e584f4def6661e5e2c458020ce80cbd39b40599fdeee78c17a136f6d5","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187098,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-22\",\"core_type\":\"Prop\",\"elaboration_status\":\"failed\",\"environment_id\":\"env-22\",\"imports\":[\"Mathlib\",\"Module22\"],\"imports_declared\":true,\"lean_version\":\"4.42.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000016\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.42.0\",\"imports\":[\"Mathlib\",\"Module22\"],\"imports_declared\":true,\"environment_id\":\"env-22\",\"context_id\":\"ctx-22\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000016\",\"core_type\":\"Prop\",\"elaboration_status\":\"failed\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.42.0, [Mathlib, Module22], Yes, env-22, ctx-22, Yes, Prop, failed, [elaboration_success], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.42.0, [Mathlib, Module22], Yes, env-22, ctx-22, Yes, Prop, failed, [elaboration_success], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0099","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b847b14e7f7a3bdd85dc70592177e20d934d5f72d38be48781763420fb76ac61","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187099,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"ctx-23\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-23\",\"imports\":[\"Mathlib\",\"Module23\"],\"imports_declared\":true,\"lean_version\":\"4.43.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000017\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.43.0\",\"imports\":[\"Mathlib\",\"Module23\"],\"imports_declared\":true,\"environment_id\":\"env-23\",\"context_id\":\"ctx-23\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000017\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.43.0, [Mathlib, Module23], Yes, env-23, ctx-23, Yes, Prop, success, [], Yes)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.43.0, [Mathlib, Module23], Yes, env-23, ctx-23, Yes, Prop, success, [], Yes)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261018-r07-lean-formal-statement-0100","batch_id":"batch-20261013-advance-20261018-r07-lean-formal-statement-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e6279333cc5e25a34c8d214cc166a94b28afefcaf5cb8bc5403b35c7edb5a9","family_id":"qf-ext18-formal-receipt-completeness","seed":202610187100,"semantic_template":"qf-ext18-formal-receipt-completeness/{\"context_id\":\"\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\",\"environment_id\":\"env-24\",\"imports\":[],\"imports_declared\":false,\"lean_version\":\"4.44.0\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000018\"}"},"fields":{"title":"Synthetic Receipt Metadata Completeness, Not Proof Acceptance","stem":"These are synthetic pipeline receipts, not actual Lean elaboration or kernel checks. The Chinese source pulse did not run Lean builds for these records. A formal statement and a checked proof are distinct states. Supplied metadata={\"lean_version\":\"4.44.0\",\"imports\":[],\"imports_declared\":false,\"environment_id\":\"env-24\",\"context_id\":\"\",\"source_sha256\":\"0000000000000000000000000000000000000000000000000000000000000018\",\"core_type\":\"Prop\",\"elaboration_status\":\"success\"}. Give (version or missing, sorted distinct imports, imports declared?, environment or missing, context or missing, lexical lowercase 64-hex source-hash shape?, core type or missing, status or missing, missing-field reasons in protocol order, metadata complete?). Imports may be explicitly empty. Hash shape is not a verified source digest.","answer":{"kind":"expression","value":"(4.44.0, [], No, env-24, missing, Yes, Prop, success, [imports_declaration, context_id], No)","explanation":"Check only the supplied lexical/presence/status conditions; complete synthetic metadata is not an actual Lean elaboration or kernel-proof receipt. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4.44.0, [], No, env-24, missing, Yes, Prop, success, [imports_declaration, context_id], No)."},"skills":["Synthetic Receipt Metadata Completeness, Not Proof Acceptance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}