{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0001","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e69ba27c291ada0a4f357bc80ea1fed9d6f1417f2d03f0df74a31306f228e72","family_id":"qf-mod38-groupaction-qualification","seed":202611071001,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(2, 2, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 2, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0002","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a4206ca2552431997f11855d60e3e9ac527d8266a4af63ceb9a3529c7caf704","family_id":"qf-mod38-groupaction-qualification","seed":202611071002,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(3, 3, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 3, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0003","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54c4586dc1467d6e7fd9c6e80be5b2dd471a48452ac141a94717d47cdaf91cd0","family_id":"qf-mod38-groupaction-qualification","seed":202611071003,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(4, 4, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 4, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0004","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d7f6fdf984bf0f4bf4e21f713ba9883e7836bd59bd7334d544344793a0ad0b","family_id":"qf-mod38-groupaction-qualification","seed":202611071004,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(5, 5, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 5, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0005","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6afd21dc4a1572dedc4754600630032b1046c863493fbabf77eee95e97ccdc09","family_id":"qf-mod38-groupaction-qualification","seed":202611071005,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(6, 6, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 6, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0006","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c7106085149a3f7a1198aa9c758a6ab5d6e256fc3e9e0fe30090b89600446be","family_id":"qf-mod38-groupaction-qualification","seed":202611071006,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(2, 3, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 3, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0007","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f9159086207ba402435a1434484b832c96c731ceda9f98510b4c57d4f54a3e","family_id":"qf-mod38-groupaction-qualification","seed":202611071007,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(3, 4, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 4, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0008","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc0053a43bb006efc08aa55ab7b3361b46c72bdc156c06f03f37cdadb89275e7","family_id":"qf-mod38-groupaction-qualification","seed":202611071008,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(4, 5, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 5, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0009","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd75a86933fd1f2ab774d6f0955d3ea4f6c32af2cb74b0263fc862058f81b076","family_id":"qf-mod38-groupaction-qualification","seed":202611071009,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(5, 6, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 6, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0010","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9fef434ed69f67413de8f666d24746c29aac70849d4f0f9e397733f3403729","family_id":"qf-mod38-groupaction-qualification","seed":202611071010,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(6, 7, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 7, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0011","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f558e659bbec403a91bab4364816b7776c6f67857a70bae77609ea563de4ac6","family_id":"qf-mod38-groupaction-qualification","seed":202611071011,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(2, 4, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 4, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0012","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26259951d4df6aa7b9dd4b898f94b8e412a3a9f9833964615c0b790f0160f2cf","family_id":"qf-mod38-groupaction-qualification","seed":202611071012,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(3, 5, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 5, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0013","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878f0817c6e103267aa4e573d1e9723533559571fbf0460d19cba757429ec630","family_id":"qf-mod38-groupaction-qualification","seed":202611071013,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(4, 6, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 6, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0014","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e35bda1053355b8a87cde76a1f9244ea5fc896419ddafa3e2b85bab9e8bb72","family_id":"qf-mod38-groupaction-qualification","seed":202611071014,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(5, 7, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 7, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0015","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bc49083b6d40ca56e78d4fce25fc6faa9dc5a5e3f1823ac609330f942ed7bf8","family_id":"qf-mod38-groupaction-qualification","seed":202611071015,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(6, 8, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 8, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0016","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0e75f7e7a5b59df9eda539138b9b617068f424b3115fe3179b256af296faf1","family_id":"qf-mod38-groupaction-qualification","seed":202611071016,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(2, 5, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 5, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0017","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c501ef80cf1f345101dfe98d6fe5b5340261aaf7751432d0d26b2536d03e7a7","family_id":"qf-mod38-groupaction-qualification","seed":202611071017,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(3, 6, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 6, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0018","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78973b6a78773d210f43ec46737f15f2322e55e72f4df9eaff78a2edbf79c3d0","family_id":"qf-mod38-groupaction-qualification","seed":202611071018,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(4, 7, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 7, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0019","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d07df2e9965e644f572456214734cd45b9fe08e7ffd218a9b136a36b07fe4af","family_id":"qf-mod38-groupaction-qualification","seed":202611071019,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(5, 8, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 8, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0020","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef0acdb1fd3c7ea1244b6ac524ae48db1153aac8e1418add6eec98ed881efeee","family_id":"qf-mod38-groupaction-qualification","seed":202611071020,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(6, 9, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 9, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0021","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1494b79c69f13bb5740ada29fc0652a1038e848810474457aa15fa5004ca2741","family_id":"qf-mod38-groupaction-qualification","seed":202611071021,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(2, 6, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 6, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0022","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf27ddeaccc7940cc9f4b387b6c91d13193a9c7b4df55072b79bd5ab5bb87b9","family_id":"qf-mod38-groupaction-qualification","seed":202611071022,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(3, 7, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 7, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0023","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8769951fb66a2a7c038c19a9e9345b94509d41297223dc71ff92732aa9774170","family_id":"qf-mod38-groupaction-qualification","seed":202611071023,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(4, 8, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 8, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0024","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"265f3fd63318140ca690452594b68ba159c8de3dfbc9b251d87bd0ffd8407335","family_id":"qf-mod38-groupaction-qualification","seed":202611071024,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(5, 9, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 9, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0025","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64f679468be8f52f238574642253e544f8a44e9d444869a5b3efbf9fd9d09489","family_id":"qf-mod38-groupaction-qualification","seed":202611071025,"semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10}"},"fields":{"title":"A Group Action Needs Valid Group, Identity and Compatibility Together","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).","answer":{"kind":"expression","value":"(6, 10, Yes, Yes, Yes, Yes, Yes)","explanation":"Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 10, Yes, Yes, Yes, Yes, Yes)."},"skills":["A Group Action Needs Valid Group, Identity and Compatibility Together"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0026","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"162cbd91355d9ab7cef45b3c92b24946bd00d552936a2dfe21cd3bf876f630a1","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071026,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0027","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80d7e58ca60c0b4c15a3bc9ccb5434342a676ac916e2c41e3453039c4d5c970e","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071027,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0028","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c54e775076b4fa73fdc588d0ed4329eb67436717e0f18b460fa757a32f9eca9e","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071028,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0029","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c468eeb5bded1a77479dbd89195f1d3fead1666cd5b00789247419517b0a702b","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071029,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0030","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"650c641507f189cdfd01a8519b709a5e7f90e13e58722d0f5b98bc5433899068","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071030,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0031","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bd2e81eac3d871ba9f82515d43bf8ec8bf4ba5da829601f68fb587931ebc39c","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071031,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0032","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9135fd5d2cda3aa0c3cb30e1ddd8d3c3b48838582427a02cdd595692935692b4","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071032,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0033","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fb26a05e774553aa73a9e99c6e7b906ce1d81dc5769a23b250052096b1ca9cc","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071033,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0034","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e2282dcebbe7240bc2495c6376d21acf125547ced9b18235d514a19ba56ec85","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071034,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0035","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a345be8f868a20dc81bc433fab10437b40bfe0fdafaa737c03d63176101c0296","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071035,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0036","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cef1050b8134023ff27a01c1fae7f846130c8f610d77779deb527bb67c376f3d","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071036,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0037","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"615e82bc1660cfacddd7c379d46644b7d45d322a39871e2dd0558d171df31a49","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071037,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0038","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b7e7361bdf19a5f96577fbc29a98263377808b8bfc46e6e4c74bf955bea03a0","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071038,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0039","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64c206bea8363e5ab454522033ce0e0eb332e979cd32407f3024b5a7caeeece1","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071039,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0040","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e86c8ab666b50e220849d369a2996bfeb885093c1f0ef6bc38d0ca8e50b07437","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071040,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0041","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be45057e47bbbcf6bb593003d39399ecb219405bedc300471d0796d391caefc0","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071041,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0042","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4243554c1edf9b49be4dd307f0e192673dfd088f8b10464fe347bbff8aceeced","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071042,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0043","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5499231848e6a3c5c090ca060af7158fb6088aef2b9297c03454761091e6c31","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071043,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0044","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe35ba957e886cbcbb1d3cddc868c36ed62d8a571549fae42223660b7297422f","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071044,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0045","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f357c96bb33c5d9929a3eadcfed60a19d1003a3945e4d1e8635daff80daab36a","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071045,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0046","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37d97fbd98287491cae9cce11042107d06e4dbec434735268284f384accddc49","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071046,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0047","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce92d186d6875f8101ed716b69c1811eb3caff975206ff726ccd16dacdfb8dd1","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071047,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0048","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f885779cd6c60e5104567c8ad0614f5e7eed7516a1ae65754c3e96a6dc0da2","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071048,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0049","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc0c8811f3ea2452642946d495454ac80b20fe8f41fac9659739139bbcdeaae0","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071049,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0050","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa6126da88c1445fd2d41c0d8bad2bb0fbedb0e2c9227afb34f4ba1834afc888","family_id":"qf-mod38-groupaction-identity-audit","seed":202611071050,"semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10}"},"fields":{"title":"Every Point Must Be Fixed by the Declared Identity Row","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).","answer":{"kind":"expression","value":"(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes], [9, 9, Yes]], Yes)","explanation":"List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes], [9, 9, Yes]], Yes)."},"skills":["Every Point Must Be Fixed by the Declared Identity Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0051","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16f63ece74a37eb4694f02899d932dadd6dab464c9718c600c7c5ce8068de43c","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071051,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(8, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (8, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0052","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d00a5344fccebc3746ececb759d03f7cd00a48a4722ed0567b26bb141afe3ae","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071052,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(27, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (27, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0053","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4efa2693e365890ad661423629825af975fb12aae4715ac9db97b50fe11f0543","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071053,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(64, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (64, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0054","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd24bd829e1cf2dc852841ec33f14c183037e4ecdaa8ff10002089383b3ff260","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071054,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(125, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (125, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0055","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e9324adf20ad3e8cebee438e5f7ac91e99d0a9dbcdf828819f27875c7189154","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071055,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(216, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (216, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0056","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0f3110548bf9c28f4b969adc085577e672a124e86fe575e766650e29c206946","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071056,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(12, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (12, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0057","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33f854161a6c5ddea6936fd17df5263c7fd277a31d26485fb0e296ab7b992a19","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071057,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(36, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (36, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0058","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33730f434053f83bc9547f857b6455850cd3ff188400f5a11c423b525a9481b8","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071058,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(80, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (80, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0059","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb25313fcf901f4fa6dc63be0e8e32e225feb49546ea87a48b2782442f59e622","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071059,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(150, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (150, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0060","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"132c7b233a7b7cac7defb7ab1220b63e284c11df345da55401d43e9aedf334dc","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071060,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(252, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (252, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0061","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68a98c6a75ada81d3e90301e5b907a2704fbd5812c4ba6cb1f3afeeea8d2400e","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071061,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(16, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (16, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0062","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6ae24bdd7aed679d30b6368641a84496b93f357045f391b7400486f30f92773","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071062,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(45, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (45, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0063","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb59d3cf0de42f42c4bafb4b1283143093f47a115928c39ea3f34d18266fc56a","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071063,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(96, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (96, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0064","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99a4c310b37def544f3573c744f222d7a25612bdd2b74b5a4e259547f297006f","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071064,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(175, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (175, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0065","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfea19cb06f20334ba736367c8d57b3c61cf3d5ba5e50206de8c402c7d1f23cd","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071065,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(288, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (288, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0066","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2302718afc163853d8ad1bff64521c415c4907549a3302cdb630c1d2e599a6a","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071066,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(20, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (20, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0067","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0031ac6574b062d62461d86b12ddf69adf08e4294aa11eda8a84ade46abb301a","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071067,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(54, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (54, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0068","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64d262a49a2609930ad12daefc6fd1e7c6080bacbb59943ae1b65661af9249c5","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071068,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(112, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (112, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0069","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bb02d5505acb76e7b8b90978bbeb8a85addd66c924ffe99a30e6b729f1564ba","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071069,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(200, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (200, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0070","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4397d6a35fb55ab4dc11c6de59c62a229efb6b3a6cc8025d4b2c74e77c23507e","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071070,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(324, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (324, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0071","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1621a6b73c05da9a5980128184b974e3da6dbaf6ae87815beb5b0eb32de66bd0","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071071,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(24, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (24, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0072","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a3920386803535861ffba17441e323cbbadcae8a4c94de0811e44660e160cc9","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071072,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(63, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (63, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0073","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3861268e1202d0e44374aeb152c320365abee4e2535aaa7a140fb0bcb0312aa","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071073,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(128, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (128, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0074","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72cc722ed6d8bcd72d0c1f13673e23ea83210cd23bf8d82e539083f3e6a7684","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071074,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(225, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (225, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0075","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0771999e07daeebdee2efd997f062d275bfacd9eac4f651ed544981e2e4752ab","family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071075,"semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10}"},"fields":{"title":"All Group-Group-Point Triples Must Satisfy the Same Action Law","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).","answer":{"kind":"expression","value":"(360, null, Yes, all_g_h_x_checked)","explanation":"Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (360, null, Yes, all_g_h_x_checked)."},"skills":["All Group-Group-Point Triples Must Satisfy the Same Action Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0076","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bc7d90f5e64c7b8a6bcd2e721befc23dc527c1406e4aa99d3ff0073651f02bc","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071076,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0077","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f97dfb72c019d8d55eca1d56c70c837373d015643d5b1770777dc8a9d4eec3d","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071077,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0078","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"524e6ce2701e1ad72dd950db8c45970add022e6e5192c5fd9a86556f71cf601c","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071078,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0079","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4075edbc84e43b9417635005c98b041de960c86727ff835f958799f1847251a","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071079,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0080","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d1106d55f24d2202ece4e1b051d0366245d12fe071058d9bef417ab6cb884e6","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071080,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0081","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15283c6635d69ca13fe8f22bd34ce4197b2ac68f35f14118978e7dda6bee1864","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071081,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0082","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b05e4a3a77200dc3f4deac920a9ddc3d4c926a6bfe15ae68161f4e3ab5f9159","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071082,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0083","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83835bd0145ac4d13941de1607109a6a19d11a03bf561c3468bec604f797dabb","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071083,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0084","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61d78b60d79eae1ec5a972bc2b8700d9267e5997d746c3e8f9762490c789bc70","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071084,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0085","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08f2f446668aef9554ce15d787df04ccafbbc05b3a29d3df8278650f3412bbc5","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071085,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0086","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37d1392db8048bb6777c9adae1092d6bb8f1d26c94539e2a74d9bfe1ac41145b","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071086,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0087","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14c85ab657829b85edc4f48faa65d41e1bd276ce9a05e8716a4af0e83e5301f6","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071087,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0088","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23e5c9e6fbc398ad88d4f4911c43f86439e582e7a1f1edb0510251b090fe3ae5","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071088,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0089","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c4dbbdad1dfb2c1d62c8d6ce89c7c6286767b977e16db889105bb6ce351a2f4","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071089,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0090","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bda7e4092c3cfb21b8101f0a39ecde1f23518ce848d4b234818b589579d9466","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071090,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0091","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d7edc3f2c00af42d76f1227d98cc7bf1d6d3d63762307b628a43b005f8066a5","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071091,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0092","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89219739c3560e48e0b8195c353f8da2d6cdcd0331c3547b81998ebb1df0c1b8","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071092,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0093","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d91c92c59d2b5afc151b452a2eb2c57f4cfb79cef926a1b419f54560a3d5326e","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071093,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0094","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae98066b3ac8a60ebdac6395fdde8ca50b92e5654fdceb413086d3331fa741ac","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071094,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0095","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c413f24234ff2b9571110858a95f16f315474921aa40a31617e5f6325d0c34d4","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071095,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0096","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61d92ba0ac08ae8accd31bf3076805396d17a6f60d3cacd60ac13cfc60b1e66f","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071096,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0097","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1038b3cde4c65a8d0ebeb66cfc90e13cbaa92343a32f9c1bbcff70026f037c8","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071097,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0098","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb75580e12c81e92a78f98156574249e0477d24ff3b58123fcc7ee4611a91b1c","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071098,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0099","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddcd676dade710230629dea3ea611bd3a6d792816e4864e7a89e8e6893549821","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071099,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r01-finite-group-action-0100","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d50db1282eb692c85de742403dac8d5a1fea5e79be8d391624afbbbb3253c47a","family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071100,"semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10}"},"fields":{"title":"A Valid Action Does Not Automatically Certify Stronger Action Properties","stem":"Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation":"Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)."},"skills":["A Valid Action Does Not Automatically Certify Stronger Action Properties"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}