{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0001","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"097364a4e8848a68914de2e1cb92c1780c290b000957f8579eb8b5191bc74839","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267001,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,1],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [0], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0002","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47a02b66a1233c490c9065e814c35363348f50c8025bae2d747ea9e00072bf0c","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267002,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [1], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0003","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8797c499b1565d37b2da031db5ec0d0a06c2769091469ca4e99b5217ec03aa52","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267003,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,1],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [2], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0004","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"266f0d72f68367a73d392fef9f75baf6bf0a6635840ed93a21e64836a58e5e58","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267004,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [3], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0005","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba472288f077d264b64c46785852029c676f5f48a351a78bd7b5bd89d97f48f","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267005,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,1],[2,2],[3,3],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [4], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0006","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595aa9aade8df061f454d4041c297648890527c6b1870dd6d452cb1d77d4b601","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267006,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[2,2],[3,3],[4,4]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [5], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0007","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"491ef10fae1de46b27b3122a1958d6422d3f5ffa40159768bfebf5ad821773ab","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267007,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,1],[0,2],[0,3],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [0], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0008","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2d8ccead1ce4d878244256cdc168c693817be2f636635dc92276ae323c03b44","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267008,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [1], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0009","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"722e3c2ee423f870c05dc377ab1f3e30e0c0f7ec83c3b757d0c467e4b39a5d34","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267009,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[1,1],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [2], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0010","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1175783c32e67dfe7a495d6b4fd5c1dc997a712635de2abda53dbe0f815aee95","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267010,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[1,1],[2,2],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [3], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0011","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8403ad873eae012ad228a82eb9db831bdb46a1081418e1abe7147673b7517204","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267011,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[1,1],[2,2],[3,3],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [4], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0012","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dd1e64dc8362b91214e2c7844f9d9cb9f183b857e19c82bf2681fdc3dd56f6d","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267012,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [5], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0013","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e696e6d054e684d0b54806c3e063a59e3a9ad3a73adb6da7163f59a12913e49","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267013,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,1],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [0], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0014","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7bd7961e14eaa413b4fdfb2bbb7db3113d5888659b1d09fe7ed54d14105d1b8","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267014,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [1], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0015","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c88f999a5cfc99cbd8b73a2b3242a36eae651b18a99dcdba3a357e1fe326e7","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267015,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[1,1],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [2], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0016","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c49786b8181ac9b4f1595fd02f1303cfacc48938b697abd05b79056e48f5c32a","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267016,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,5],[1,1],[2,2],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [3], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0017","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d269e62d9a555005fbd98d98f58781419d7ce9f4734a464946821c5ca03ebe0","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267017,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,5],[1,1],[2,2],[3,3],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [4], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0018","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b57ab0f7ab700c2df2b8d78e13d65874ae1ca58cd22ab7548b4c3e9b8415d258","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267018,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [5], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0019","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea2dba917462022e9bc93ec66ed77006b9dc2c1616f0e897fb61cb0b0222a5d7","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267019,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,1],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [0], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0020","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab722bead57b3f743a6c5c2d5b6d06e2c6d844f51b2f2f9d6b239d8038c269d","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267020,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,5],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [1], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0021","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2539df5a28e2ce7c7a683f4a70955a2ab49a9807cd9e44f340cbab0e9b1de464","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267021,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,5],[1,1],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [2], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0022","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c823c053387d346c5c2dceabee3c65b40c0fd432fcd1779ef924a5c7c056cee2","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267022,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,5],[1,1],[2,2],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [3], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0023","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2f5f14f582464ae99bb24b13fa048a12fbff7b8ef6250d0387a4ce47903b5e4","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267023,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,5],[1,1],[2,2],[3,3],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [4], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0024","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93fcb18bdfd88cc423dbca34aa1ce51146f98e4285445f401fa0b8d5c0ab424a","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267024,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [5], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0025","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5299925afc6c5e524507bd8560dd97f691e878e3cf3978c1ea82147037eaace6","family_id":"qf-mod26-equiv-reflexivity-audit","seed":202610267025,"semantic_template":"qf-mod26-equiv-reflexivity-audit/{\"n\":6,\"pairs\":[[0,1],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Every Carrier Element Needs Its Diagonal Pair","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).","answer":{"kind":"expression","value":"(6, [0], No)","explanation":"Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No)."},"skills":["Every Carrier Element Needs Its Diagonal Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0026","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb41fbf7e6ab04b71c237758791791fa0386ac347b842f9cc56cb00b3ab6a4c","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267026,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0027","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32baca7814f4ea224dfcf11f63847d1568108650409acfac6f02e77211887262","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267027,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0028","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dac2923bdad962d1bb4c066ca4b2a88d8a37d1c3299879f4a468836a104ad35","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267028,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,1],[2,0],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0029","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dc498665f46311ec64234c5a0250f73e460b14033e16ad71180e645879e70f1","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267029,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0030","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17d905ee4329576bb4dd781a206e714fa17aebb3a240d427a62c0868afe6fea3","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267030,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0031","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf4828a8db5ce4e437ab80bb28fff2a2ab709f68e650d5b6954413c3b1575f1","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267031,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0032","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b5a6f263bc47d7d5a7a47e7e9af8f19353293ed2a0a0d3448261307cc2d348","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267032,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0033","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35cc0ae69b113605ab3c4d6e57173d19f30b8f7ba48a478969ff8bb62b268c1d","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267033,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 4]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0034","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b8e5f8e83b2eafa9d5f290cd8aea57cba5ef5d0156a63912d4b679cffbaac2","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267034,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0035","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcba927bfe3f0db39c31e60d502b41fbf6f511fff93a05760813eb6599787880","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267035,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[1,1],[2,0],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 4]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0036","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"860ab241e6074faf1748c2e0a00a3f6c8f23af8f96a093e342be54e2273074d3","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267036,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0037","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92682a91589adec97f33095f3e08b6071a69c11b56c06ceaabd73ffde17e20a9","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267037,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 3]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0038","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42869a5ef4ad313f9b9e00c0d00c53e3aa9aabb4f7e584569be9b7e61c36dde1","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267038,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 3], [0, 4]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 4]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0039","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d80b93f1127d27b355be9cc96d91932987bfc42a348279cbd4a89e53f34f0088","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267039,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0040","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"276fe52d6ff32574592d8f484027eb4ecffad7069900b962b0fa1a9922872c3d","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267040,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[1,1],[2,0],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0041","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd60095ee46a3d7e08a536cdd26a77464059a72fb0aa3c6a0f4c448e73be8c33","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267041,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0042","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e08804983f115509151ad6f6e798dfdc9f7d8b6c425637a66b7864990c2958d6","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267042,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0043","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09739f22e2546682e52723af7a6aa7d6f1a407a839f63116f2c298fb51b925a9","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267043,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0044","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32a24659c9bd1bf64de6214df3a22a07a73d7e776266b250c6ea38693cda040e","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267044,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0045","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"417c8833a2400cc78c8736110f5fdf07e3f8f2170445d07ebda87a7467b4089d","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267045,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 3], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0046","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07cb66d80d4635308ee2a3b68527acafacb027745f39790a280e8f93df81757f","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267046,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,5],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0047","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3abab6c0466da3539143ee917da31f9451df140f5fcf8fc45061a4bfb7860c46","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267047,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,5],[1,1],[2,0],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 3], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0048","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d19f6837a451665eba4bc4a9d4a7aeac0ab6adb6223f91862a8fdf5920412e72","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267048,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,5],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0049","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29daa3c32cbb90754f1edec22ec5ed04fe682a9d235897af5159440e761cf73e","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267049,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 4]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0050","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f8c2a91a360ddf4f0564fb01f7884e8ff7f511358b76d9575ca513a6cd76739","family_id":"qf-mod26-equiv-symmetry-audit","seed":202610267050,"semantic_template":"qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Original Pairs Whose Reverse Is Missing","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.","answer":{"kind":"expression","value":"(6, [[0, 4], [0, 5]], No)","explanation":"For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4], [0, 5]], No)."},"skills":["Record Original Pairs Whose Reverse Is Missing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0051","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51338f8288518e3290c221c89b68bfaa6380e544def3f447a6aab8a6d974aa7b","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267051,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0052","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b94054615a98f0116bd63f1f3e5986d6c10717ce5f69923b487635fb295636","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267052,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0053","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc9d9aa5d37e12f8ec8ea0be4b88fc19203c623dffeb59f7d26f9e2ad551151","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267053,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0054","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34fd07a0e65dafd47412a99efaf90f08116659d4b4a23c82187f6cede61baa56","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267054,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0055","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe57ce2f4b6d62879ceb799ae3eaabedf5d97cc7d8179f2599d66bcf2e5e1957","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267055,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 3]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0056","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9bb2ebd077e461c22129c46cacf98400a7637cf2b5730c76b5aaaeb77cd0191","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267056,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[3, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0057","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12d40b5a0d89b545df1f29d86fe92d59ad592ba41c0e802c3b34d1ffb22812db","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267057,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[1,0],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2], [1, 0, 3], [3, 0, 1], [3, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [3, 0, 1], [3, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0058","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"856cf8ef619ff158bfc5ff3e8f85c563ff1e35ac157272ea961d8f782a1d6d1d","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267058,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0059","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f589ae1bee3eda97d7d36e6471c02e5126b32b7c05187f9a6eb361bcb6d2a2e","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267059,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 4]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 4]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0060","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1befe92813797b764e83f25754f706da53f2fbce5f3d092e098f1b42fbf58d82","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267060,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[4, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[4, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0061","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6116b33455a9667963c5a5aac0c130bcd6cd014fba1ac72a1c790f122f1d99e9","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267061,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[1,0],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4], [1, 0], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2], [1, 0, 4], [4, 0, 1], [4, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 4], [4, 0, 1], [4, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0062","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"202b3961a462b56e75ff1ad2b2074a34279d3feeabf169ba0be30182cd757185","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267062,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[4, 0, 3]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[4, 0, 3]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0063","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd00618f55a41802c2608ff4dc89e8e19fb33c52f6cf2db826e78ed299db0662","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267063,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[1,0],[1,1],[2,2],[3,3],[4,0],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4], [1, 0], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 3], [1, 0, 4], [4, 0, 1], [4, 0, 3]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3], [1, 0, 4], [4, 0, 1], [4, 0, 3]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0064","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544712b29d4979f8dfb9ec754c79b9eb8cd5ca1c3a64d80d279fe4cadb2b05ae","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267064,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[3, 0, 2], [3, 0, 4]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2], [3, 0, 4]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0065","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4597e277b96fde62742cb1e2e43e70d4bb8ede6edecc44d412cf4fe98fd3290","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267065,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[1,0],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2], [1, 0, 3], [1, 0, 4], [3, 0, 1], [3, 0, 2], [3, 0, 4]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [1, 0, 4], [3, 0, 1], [3, 0, 2], [3, 0, 4]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0066","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e19a301e9a09204f63d4cb05d3118d396e3151f9ea5e0a7980929bccd9a5bf9","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267066,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [], Yes)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0067","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8341e7bf48ad5876a3bc05fa98265b81c98f1953c4692ee65fa26bbdbbdfc999","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267067,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 5]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 5]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0068","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cdd7e1617c2616773620ea2c2a3f2087625317a03d71b48b45708aa0438dff4","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267068,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[5, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0069","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"838e0e578e11b3f6d099b791ecd196788fe568478465bc925c243b1e24f2868c","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267069,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2], [1, 0, 5], [5, 0, 1], [5, 0, 2]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 5], [5, 0, 1], [5, 0, 2]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0070","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edcc54f9c82fc2ec81dcd5d3bb8de880d5136dfb88a68cff512ac0a91b28791c","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267070,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[5, 0, 3]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 3]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0071","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f40135c33c59fc78adec32101961c4a9f4aba53d31042039ccdca5e75691b46f","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267071,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 3], [1, 0, 5], [5, 0, 1], [5, 0, 3]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3], [1, 0, 5], [5, 0, 1], [5, 0, 3]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0072","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6249da734977556b85551f82ab8dbe9bdf989f101fd81d29d78d1f26ef3d8d84","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267072,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,5],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[3, 0, 2], [3, 0, 5]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2], [3, 0, 5]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0073","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce3941c6577c679594cb5c206a6bd43e695d0fdbbb3d4d762d4d60aea91fc4c7","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267073,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,5],[1,0],[1,1],[2,2],[3,0],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 2], [1, 0, 3], [1, 0, 5], [3, 0, 1], [3, 0, 2], [3, 0, 5]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [1, 0, 5], [3, 0, 1], [3, 0, 2], [3, 0, 5]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0074","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9e84807361d077be57e153ac65e9859e468f143ae249bdc46788051ec3dc272","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267074,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[5, 0, 4]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 4]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0075","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f4cc037b26c94d4d9d89ead81da4f273284427dabb3d3eab8874e47553043ea","family_id":"qf-mod26-equiv-transitivity-audit","seed":202610267075,"semantic_template":"qf-mod26-equiv-transitivity-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[2,2],[3,3],[4,4],[5,0],[5,5]]}"},"fields":{"title":"Record Every Failed Composable-Pair Triple","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.","answer":{"kind":"expression","value":"(6, [[1, 0, 4], [1, 0, 5], [5, 0, 1], [5, 0, 4]], No)","explanation":"Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 4], [1, 0, 5], [5, 0, 1], [5, 0, 4]], No)."},"skills":["Record Every Failed Composable-Pair Triple"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0076","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4524c5aa04433ddba532510906afdd247d320aa25361411a4065a228cd56ca7","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267076,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [], [], Yes, [[0], [1], [2], [3], [4], [5]])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0], [1], [2], [3], [4], [5]])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0077","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcbeef2e0f2ccb0c0a7b00ea48e30c9b10645c9d4f63bd8d5bba8ce402f92c18","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267077,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [], [], Yes, [[0, 1, 2, 3, 4, 5]])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1, 2, 3, 4, 5]])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0078","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ff697dde27aa321f487608365576cc0a94dbeea4d5506761cf275829738353a","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267078,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[3,2],[3,3],[4,4],[4,5],[5,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [3, 2], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [], [], Yes, [[0, 1], [2, 3], [4, 5]])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1], [2, 3], [4, 5]])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0079","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fafb9ab74edaf3c26830e3157213fee6e92f4f36da919ba9180843b8dc9462ba","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267079,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [], [], Yes, [[0, 1, 2], [3, 4, 5]])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1, 2], [3, 4, 5]])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0080","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d7a41f715d40b1ea9a4f5296afb4289e151297d39a764325406ba7bd5d02cd9","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267080,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[2,0],[2,2],[3,1],[3,3],[4,4],[4,5],[5,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [], [], Yes, [[0, 2], [1, 3], [4, 5]])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 2], [1, 3], [4, 5]])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0081","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a628c69d44c5296dfff00a58a4e2b8be08c1cfe3acfa033c9d3049878e339325","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267081,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 2], [0, 3]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0082","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"328f8d6566de00e6a2c82baf42a4917bd56164744f75c3de04485c6395828874","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267082,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 2], [0, 3]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0083","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4f3d27c89564058916351670a28829c9b5818c5c6e3faf8d4fcc9d18bf5028b","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267083,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0084","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"675faeb5d6be9cda75eb870875d779116ac6b0f49845f6b9dbe604c19b3d9e76","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267084,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0085","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"994ae2f61a36f00ff423a1d4da7e81b62869773585c76ff7df6d395d338829f9","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267085,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 2], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0086","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31b53f424a14e1cc1f2b56e4637cb838e1d25ca461bb9b2c37d3bd32a655f0f4","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267086,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 2], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0087","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b2537707fc90caf4c0e07b28387381b46aa82d36944013da4a34f7bc523d84","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267087,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 3], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 3], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0088","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d200fc8da778b4e095c4d5d56c63cf09b5c91b0fa7cc2065c0037cc58d67cb28","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267088,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 3], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 3], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0089","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2bd83dfb65f1ddc8841f5e823a8df049035c37f066d80d10503b7aebfd2cb91","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267089,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 2], [0, 3], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0090","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76cbded8ca0cb2e2a5b522bd2d414c8ec9330dd9c3f4cd19e044046ec225c4e0","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267090,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 2], [0, 3], [0, 4]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3], [0, 4]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0091","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7293cce29b0d676b051236918c855f3ad6300aac6ad04f44c9a89c59efcfff68","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267091,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0092","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d898471b9913049a591e7677e10f499eff3877f6fa257a838e246c86a6fa42","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267092,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0093","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f05f67e719f52278ff5fa7338cdc915810f050127803650fff96de62df27c020","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267093,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 2], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0094","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"325907461a88788f1cb1b6877b2fa6d9c5e45819e1c78ad21a05edbc367e0393","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267094,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 2], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0095","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44b4d7ffe44d4e08482c69903371100e7b33b792940c12ad47c41f518f2bd8e2","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267095,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 3], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 3], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0096","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46800008481adad495630564c84dcba64cca9ec2a8eb932c3a0522dc95fddbe4","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267096,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 3], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 3], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0097","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63203140810e78e8f315c4d9fb836b3044e61790dfcdd24f6e2ffc620639898d","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267097,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 2], [0, 3], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0098","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48b19341f78b6ae9b0656fc02c3edd2faa7d71fa9f44ce1425cc699d59915587","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267098,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 2], [0, 3], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0099","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e690a69ef7b2ac851a385e17819241adea072a7996305b0f0af7b3d58cea01","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267099,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 4], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 4], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r07-equivalence-relation-0100","batch_id":"batch-20261013-advance-20261026-r07-equivalence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b76f39ad171578e449d08bff810579af3900c9f0f172a3c33f55a7238d70632d","family_id":"qf-mod26-equiv-qualification-classes","seed":202610267100,"semantic_template":"qf-mod26-equiv-qualification-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,1],[2,2],[3,3],[4,4],[5,5]]}"},"fields":{"title":"Emit Classes Only after All Three Axioms Qualify","stem":"Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.","answer":{"kind":"expression","value":"(6, [], [[0, 1], [0, 4], [0, 5]], [], No, [])","explanation":"Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 4], [0, 5]], [], No, [])."},"skills":["Emit Classes Only after All Three Axioms Qualify"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}