{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r06-maximum-matching-0001","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c01b8abe91942f704927b35d8f606c650002e994a6c53969bf977b9f2f3fffa","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036001,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(4, 2, [[0, 1]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (4, 2, [[0, 1]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0002","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baac79700ff6d6d921c93b7c45fc1e72b75882587ab4b11369206897e4619b54","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036002,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(5, 2, [[0, 2]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0003","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03d460d0a4fc92d02cf555ea077b76cbfa18d99de1f3850a1ac1d6bdfbcf3818","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036003,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(5, 2, [[0, 2]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0004","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e1a56397330a25b9bcc8439f7555b6ff22ef7912fbd46e66948a2531f0c0ba","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036004,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(6, 2, [[0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0005","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb11c5f28104e5368632a79b4765b7e28d4fe2060bd03edae39c8d1de2336a8","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036005,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(5, 2, [[0, 2]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0006","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57dad9a2385422b234977785008c5d03f96305de688b41edfdc6416865c58955","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036006,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(6, 2, [[0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0007","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2c37538c2bc91b20332560eb82383a2b7ab7395f0e45ed1af5fd07a4fdcc0d4","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036007,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0008","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb665956228c7efcbbd42d762be8152ba7e9ae76bd650f6a0269e8b6e18da860","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036008,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(8, 2, [[2, 3], [0, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[2, 3], [0, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0009","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66102b1f007c1d353502a44cfb3a6dce6ebe8df68254e0244b325dfde6f240a6","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036009,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(5, 2, [[0, 2]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0010","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db4d80842e1b0773545603af43385538a09c86d92b99457b025b5540c2c7ba72","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036010,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0011","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"269d08bab2d8446f64a6b5f3efc7ae78af8a2b94be8a1a7a9a1bcafe28aa370e","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036011,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(6, 2, [[0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0012","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b522260a65806d54970bab19c726179686003cb32101a08be3fc08b6b736b2e2","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036012,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0013","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0be8dcd1738285cfc490bbd0918470dd53d822f7e380ae674623dbd5483e31b","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036013,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(6, 2, [[0, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0014","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ef1c37d52093fe0253646b8f1eae2994c7ba29a306afbae1718fdb6d928323d","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036014,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0015","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c8c46dcdca566e1b8f543827299081b1734cf96ded5d31396b1712a80583ca7","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036015,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(8, 2, [[1, 2], [0, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 2], [0, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0016","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7a0856538f26b6112fe09a3f2c6627c5cad424243776da2d3858ffe67361f9e","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036016,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(10, 2, [[2, 3], [1, 4], [0, 5]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 2, [[2, 3], [1, 4], [0, 5]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0017","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"940f8620d7ada600824c560fc5facd61e18e032c0454843478e27ec01d1fcc18","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036017,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(8, 3, [[0, 1, 2]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 3, [[0, 1, 2]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0018","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa3e541832f552ecb3406dbb5ca2643f4a48bfc84e7d63f254b6244126dae2c","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036018,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0019","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb6d3d69f3339e37c773c126ba206306f541e68e2be6f260083e3eca174bf760","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036019,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0020","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1cac9dd7a4a1e4dc469733b7901a43c95df40866dd0df4aec8552aabf59c513","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036020,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0021","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33c6d2850e631ed1ded50129a6524d5fa1c26af11f869de050c959082ff19903","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036021,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0022","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd69aae581fc612c7041497fd31f4fba3014f5b18387831c3068a1153f83d1c9","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036022,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0023","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4892e52f1b602d62565e4598502b7649fb00e2998b0d54a790bbddf97fd22376","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036023,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(14, 3, [[1, 2, 4], [0, 3, 4]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (14, 3, [[1, 2, 4], [0, 3, 4]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0024","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85342826462dd790f3f1ad2b0b2b6f45f8da8fb2ea99a3132aa5e494e9be2535","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036024,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(16, 3, [[2, 3, 5], [0, 4, 5]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (16, 3, [[2, 3, 5], [0, 4, 5]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0025","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9acc986712e448821d3b688cfbc5636292b094853cada3d4cb8b985af352bacd","family_id":"qf-mod34-maxmatching-maximum-size","seed":202611036025,"semantic_template":"qf-mod34-maxmatching-maximum-size/{\"edges\":[[0,1],[1,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"The Maximum Size Requires All Valid Matchings","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).","answer":{"kind":"expression","value":"(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)","explanation":"Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)."},"skills":["The Maximum Size Requires All Valid Matchings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0026","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0efa9d453aa95999a10b2537fb16caff4e984ebc2e3770a0788ee91cf9d7424f","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036026,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], Yes, 2, 2, Yes, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], Yes, 2, 2, Yes, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0027","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c2a997e916be1a7fdde525917c706a0de4107ec591e95048aca6735f75c737e","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036027,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0028","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d84b52ff69bced269594b5f6b1365826f9e0e39337ed85fc91a9c56a8d92858","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036028,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0029","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"239b0decda52301ba2499a0d518ff9d2663a574fdfba1c1de7d150a292b7491a","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036029,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0030","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"591c86fea78132323444a0a5a2ba90ba50c3e6f5be00d403a0b2b1097252fc98","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036030,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0031","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"135ee6ca7815f0d656ea6787fff503c803db295a2c070a38575742537f789f5c","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036031,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0032","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bb18f1d26b32578d1f3ac72a7e5b9240e0077c07d15f8061f6a39c3af1ef571","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036032,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0033","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88180e0d0f04b185146212ff322a0805759510f72a235585952854d53ea71403","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036033,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0034","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d2396e8b10777340073e6629581ab8185ca2ff403c3b9c6f0464ed16b4e1e5f","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036034,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0035","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95813ab96e5e3a3ff76308c57c0e5005626434d5bd3dd1a57fe9beba21578645","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036035,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0036","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fec994dd8ecf175cbdeee936d7b082d9e8ac250059efecce6c3e928652ed1f40","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036036,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0037","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d44cdcedfacfe2d3b43e60b1448f5f3422b3bef8d3d530ee14ef9a5908646ef","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036037,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0038","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d5b90a759366a6cdc8662137f41499cec280069daf45701faaf3c0d82677800","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036038,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0039","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f0cb394e2092115f11c8b75083711dd7ca7bba37dae1553865274a657560d9f","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036039,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0040","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f76ad0e3fe1122f70e26c390b2189a388f3dc7838722f26c20b229d059b6583","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036040,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0041","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b487eee59862e88e82643f6f41ba224f9516c49a2add51a721e4c7a74c15883","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036041,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0042","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e519f910f7cf8ee88078e0e31ac5f3ce9189fd6b06cc88784fd10944b2b40010","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036042,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], Yes, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], Yes, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0043","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d8c433fdf253e465d8ead208c79038a8674ca920f0bf23cca767daf4a20774a","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036043,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0044","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc434f448851ba1c70004afb915805351eb82ce752a8c62ccc684ea6c544cf40","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036044,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0045","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"801c8e32c4b2e9e92dc7a9ab759a539d5fc1281613582f83a82e91eb53386cbc","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036045,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0046","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa98c003b1c4735d142c401808b1ec5e8d44a34cbd39d51fabe313b0b139eae","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036046,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0047","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d75de78b4b7de06fb8a19c2ac7f2ae07743ea79059b89b7aa46f4d62247d54d","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036047,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0048","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee290d38f5bd895c2326048fe35abb7f56039e3ea0d2705d945d3357b010653","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036048,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0049","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c1dbacbe1388858b0f4db8ed4c2d432887b2f05a34b33348982a58a385956a4","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036049,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0050","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b6a3a67dea626224afa423440800882cb36237daf206daaa735294ae123a1b","family_id":"qf-mod34-maxmatching-maximum-audit","seed":202611036050,"semantic_template":"qf-mod34-maxmatching-maximum-audit/{\"edges\":[[0,1],[1,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Validity Precedes a Maximum-Matching Certificate","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).","answer":{"kind":"expression","value":"([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)","explanation":"Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)."},"skills":["Validity Precedes a Maximum-Matching Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0051","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51fd2d907ab17ff881ded3331d1f560be16e818095aaefbb10e85db4f0b091e1","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036051,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 1]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 1]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0052","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e0603026074d9d0253bd1605d8a01727f639fe66c64e5a963dffb1e930c9e54","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036052,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 2]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0053","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"648cb8afd9bf064ddbeef1ae4d136584f1c503062d359c40a72e003c87247c9e","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036053,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 2]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0054","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ccc5770ec1c9ce1a1ef48937bfefd27919b2db5d06575f7f16b4d4630a50d68","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036054,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0055","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7d97bb07a7699279162252fddea122271d04a9d7e5edd7452dbcb7c7152ab1","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036055,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 2]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0056","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feb5617deb790ac1f845b2b22538e2ba24cbcec746bd64f4a1a90e8c9fc4f4cb","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036056,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0057","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2049c1637ef4164a8062c9361d071da26013b3dd0775cf34765d9fa222d14d20","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036057,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[1, 2], [0, 3]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 3]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0058","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c32d18ba21ce023cb5a643a33f0150dceac8f4cd4e66f1ef9ff75077aa7b3a5","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036058,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[2, 3], [0, 4]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[2, 3], [0, 4]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0059","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c323ef808c5a8b81cd74a48e1d545538592f9d63f520545b2c3a8b28f1400ce","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036059,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 2]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0060","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2d2e31f23f87b03cc3f81780b90bc37bddd7ac72d5bd3e2d5a778461fe9ffbf","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036060,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[1, 2], [0, 3]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 3]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0061","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27da9a3dd2b582c816dd0febfdec5a75d61a870c985f4d7a861cd7810087d004","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036061,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0062","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51441dcb111d2e6d1967fadb44560f5af3fef3752139fd8f39fc8ef3c1ad1773","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036062,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[1, 3], [0, 4]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 3], [0, 4]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0063","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f332dd1cf9f6464a9b37efd43c0725fcb11860109f096bc256630d0898e7b9c8","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036063,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[0, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0064","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9017bc040c00af5619221e5bb8a5358a9da68f3a56b429e6425affbe2ca84cbb","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036064,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[1, 3], [0, 4]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 3], [0, 4]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0065","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a4ecc2d389d08e690d15188e034076b2fbe10e83c3dd665b83a62a5c269d655","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036065,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[1, 2], [0, 4]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 4]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0066","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dca334dac76bdaf35e75d83da7a92541155a721b026c9f4b5357dab399760bd","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036066,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(2, [[2, 3], [1, 4], [0, 5]], 3, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[2, 3], [1, 4], [0, 5]], 3, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0067","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3d6b974075f8d725724a7096fdf0c05d0966ff001c8b721e12ff166217dba00","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036067,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 1, 2]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 1, 2]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0068","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edc9a671b777ae9e65adaea830546a4a21acc8cf389715cac246c23ba9fdbc6f","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036068,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 2, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0069","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ab4620b8e3263ee72d098e7937ac098dd7b1fb1e0aa2382480e4e43a846c673","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036069,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 2, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0070","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac1fbde887ba4351ae3962aa573efaae76a1e1d00d40fa98dd7cb13cfbd2dd7b","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036070,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 3, 4]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 3, 4]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0071","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d8feb6b0e378dac86ed40727a9c54b6129b93ef613e8aadcf84768a7957cd59","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036071,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 2, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0072","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd7a8083858681a7b2c09c61958df47f4769fada7c2d4d1fe5d17bbede6d9c62","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036072,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 3, 4]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 3, 4]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0073","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af694e524e74a5ba168ff4f01d96d35908334c427efbdd2242ecf99c05b5ec0","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036073,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[1, 2, 4], [0, 3, 4]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[1, 2, 4], [0, 3, 4]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0074","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8e4f48d171fefa3954eee369432862096ac360863d48164981cac3d060cb1b1","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036074,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[2, 3, 5], [0, 4, 5]], 2, Yes)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[2, 3, 5], [0, 4, 5]], 2, Yes)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0075","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"924bae10dd587e8628c03a7ca1c6cfc171ca8fc710298ca2254d77cd06334eed","family_id":"qf-mod34-maxmatching-multiple-maxima","seed":202611036075,"semantic_template":"qf-mod34-maxmatching-multiple-maxima/{\"edges\":[[0,1],[1,3],[2,3],[4,5]],\"n\":6}"},"fields":{"title":"Distinct Maximum Matchings Need Not Be Perfect","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.","answer":{"kind":"expression","value":"(3, [[0, 2, 3]], 1, No)","explanation":"Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No)."},"skills":["Distinct Maximum Matchings Need Not Be Perfect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0076","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ad5aa88582db8c2bf12620a43a2bb4afd1c80d65d43771f23507bb26f8732eb","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036076,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], Yes, [], Yes, 2, Yes, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], Yes, [], Yes, 2, Yes, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0077","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ac5b8ae4ce05c86a7af032f5a4b2e24c3423d28b5e62c3f7bb9c934814e9f9","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036077,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0078","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"112b7ad0fbde333e4b89090d28237cb73e85cd9b01747f27df42871584c43367","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036078,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0079","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4ed043a25bd15ca933e57085ae9601d4033ef836def7fc36bf7b081ce42d2e4","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036079,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0080","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcea096f8f3574637d358dcbc7522dab062f4819f6b024f591d4a320a4333408","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036080,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0081","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca343a766dc74863900943181f0a6d97bd15bed9488bdfeccae8ed551bb481e6","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036081,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0082","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e972b5ab3df41afb66f5111755d4cd49c266fa8d07ec7368dd3a91a6a748bb3","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036082,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0083","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6d166504556b2f5338e3075c7a79dfbaed3f6934c20e7bfd6e9fdd074302ba6","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036083,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0084","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6aefdfb2c2beb531b830ea71c7ef7fc21ad5c874d3b7e39b8a7fea320278942","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036084,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0085","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc7e7964e2ed23cab0f0cd78f60fe6cd4611fe35e7de1b271c141124b1c3d286","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036085,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0086","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a91349231fb14565706028c491c7da2aa64a71bffce3145a0bd149d2090a7d2a","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036086,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0087","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3895930468e19858b24c21986975d1ce3a94f98e64ce15c91a76febd17f3d9f","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036087,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0088","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1246d113d4aa8c20c37720698acc6741c3682ef993ac4f5d7e18ed08d184821","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036088,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0089","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d091e94ba35697f520eb18d174008632ea78a2ce2c257d1fa69d013c99125ab6","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036089,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0090","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5569d87db559be2b4e7ec2c6207eb64dc65159f0292f27a09cdf47b82b09851","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036090,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0091","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d4dcc6f9e7cf83fc6d49d8264e0c8f4d606eb9eb128893412214e9d00a1d9ca","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036091,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0092","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a8ac55fac899c57979ef0ef2a2fff27e1571b5fdfae3281c5c0330e4bbe0301","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036092,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], Yes, [2], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], Yes, [2], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0093","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f2295992221361488634183ad79df1e7280bc5a8ddbe92d1b7614b539eb2836","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036093,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0094","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e787a74af16e9ecd45494cfe322aaa6fe379776c2ee02acd07df57243ddfb7af","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036094,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0095","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0812953a6d28a4c44ecaf2b68c91209041b8336f71c66a98d0e8d3008a05aba3","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036095,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0096","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f66561707b6f0ba9b203a22a3e20012b08b88f84b9b1e3aa67c067e87e971b1c","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036096,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0097","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97d650be36bfeb78bd0c57a28035e595236a9710fe7aaf143fcc4b1b5b9b217c","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036097,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0098","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e8278d159b417adda01d75825445aa28bc0a4bf6ef1186d6838d042be388778","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036098,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0099","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce5c2938db84d73392c29378e597077a8a610ca17630588db64abb4fb0aa6662","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036099,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [5], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [5], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261103-r06-maximum-matching-0100","batch_id":"batch-20261014-advance-20261103-r06-maximum-matching-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bc374042f9001b4a725bd6c88e4e9a2cc0b29bce71edc23b4ad1d452734834f","family_id":"qf-mod34-maxmatching-maximal-vs-maximum","seed":202611036100,"semantic_template":"qf-mod34-maxmatching-maximal-vs-maximum/{\"edges\":[[0,1],[1,3],[2,3],[4,5]],\"matching\":[0,1],\"n\":6}"},"fields":{"title":"Conjoin Validity Separately with Maximality and Maximum Size","stem":"Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.","answer":{"kind":"expression","value":"([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)","explanation":"maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)."},"skills":["Conjoin Validity Separately with Maximality and Maximum Size"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}