{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0001","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"403ae1e6d8b62505769e22e2f84f886c4f30a5556a96fe74ab3d93fb5dff95c6","family_id":"qf-mod30-quotient-classes","seed":202610304001,"semantic_template":"qf-mod30-quotient-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":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5]], 1)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], 1)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0002","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"927fc2c9bde8f3e0521e991d42a385b13933f70e0cb7dcba4f95add4c77bf7c9","family_id":"qf-mod30-quotient-classes","seed":202610304002,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1, 2, 3, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2, 3, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0003","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1e54d536d44fc259b13ee2eeac2e1d7e33e859993d7ed7565b732b2172637bd","family_id":"qf-mod30-quotient-classes","seed":202610304003,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2, 3, 4, 5], [1]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 3, 4, 5], [1]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0004","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f37c70cdcadda8059ffe8f3dd59ba3905ad0fb834b24daebd56e92c839cbc2e","family_id":"qf-mod30-quotient-classes","seed":202610304004,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0005","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e00d2a8b10569da5a5b8537c72ac90a9765dd9bb831f1d81939e86c12d29d6","family_id":"qf-mod30-quotient-classes","seed":202610304005,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1], [2, 3, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1], [2, 3, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0006","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aaf0d648b33b11ca0f7de6f9f22d192709942a5c386ff6af7820e90470b2952","family_id":"qf-mod30-quotient-classes","seed":202610304006,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 3, 4, 5], [2]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 3, 4, 5], [2]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0007","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56928812341566ed22fe058e057dc8fb2c0fbfe7ace2793908bb5061f37264b9","family_id":"qf-mod30-quotient-classes","seed":202610304007,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0008","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ac37214af77293dcde12f2a20bda72c7fce0212ccbd5da370b17def8dc8694e","family_id":"qf-mod30-quotient-classes","seed":202610304008,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1, 3, 4, 5], [2]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 3, 4, 5], [2]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0009","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07eb1840ceb9325622807af137b009d078c28752f9493132bbd3d480758ec64b","family_id":"qf-mod30-quotient-classes","seed":202610304009,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1, 2]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1, 2]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0010","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0afe4cecf38ac97f838de8a62fc1cc99218b310bae4c8aa70fc513d95ed4176","family_id":"qf-mod30-quotient-classes","seed":202610304010,"semantic_template":"qf-mod30-quotient-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":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0011","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e30e8e8826c18434d3f7f849157d2073fea8170c2e7426e94112d33127d406","family_id":"qf-mod30-quotient-classes","seed":202610304011,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1, 2], [3, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2], [3, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0012","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80847157ce524840097cbf539c427585f0c396f7c25d286f7aa06d6bb680b6e4","family_id":"qf-mod30-quotient-classes","seed":202610304012,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1], [2]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1], [2]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0013","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e849b5b02a348218ab966bef16b9457cfbd0539f8b8e9cebad0215bdfd8ecbd3","family_id":"qf-mod30-quotient-classes","seed":202610304013,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2], [1], [3, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2], [1], [3, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0014","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"738da62f67ffe246caac54837276e9960ce7e93c418603771810efbdbf676fa5","family_id":"qf-mod30-quotient-classes","seed":202610304014,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1], [2], [3, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2], [3, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0015","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1c508408fa9845a974c30146c9e1e3aa1c6448475d803e8186dd87b3c83359f","family_id":"qf-mod30-quotient-classes","seed":202610304015,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 2, 4, 5], [3]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2, 4, 5], [3]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0016","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8e1c27df7d41a8d6d7ec551895576ec2f5f169dfe8f3fa687ba8b73e4cb7c16","family_id":"qf-mod30-quotient-classes","seed":202610304016,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0017","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31928e4796f4ae0b58ac8233f8666a9c9bf63bc66bd2f1fef09a31d17c44bd01","family_id":"qf-mod30-quotient-classes","seed":202610304017,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1, 2, 4, 5], [3]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2, 4, 5], [3]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0018","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d293453f69fcc66affbfe13ed30fc8488ab098519f7fd733bbcd5e7b95e5d9e9","family_id":"qf-mod30-quotient-classes","seed":202610304018,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1, 3]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1, 3]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0019","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c58bc1fb50c356290d520c596a30f35448b43fbadf928b96c68ec42dca8e51b","family_id":"qf-mod30-quotient-classes","seed":202610304019,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0020","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f67049a73f185b65ee8e9aa0d3f02c4f6ee5eed4165edca620c27554c6903cae","family_id":"qf-mod30-quotient-classes","seed":202610304020,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0], [1, 3], [2, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 3], [2, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0021","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb5514c0bf392a32e704810ecb0731d6f9a7d5bad3d0537e86c7f269c6ff291","family_id":"qf-mod30-quotient-classes","seed":202610304021,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1], [3]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1], [3]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0022","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aba7b2c1f3dfac9dacf1806062356f8dd8734554a1d5498516dc5dea7812576","family_id":"qf-mod30-quotient-classes","seed":202610304022,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 3], [1], [2, 4, 5]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3], [1], [2, 4, 5]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0023","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"907a881495f8aa47d00479756219662f0409692b15bc549bc1a8854a5b8806b9","family_id":"qf-mod30-quotient-classes","seed":202610304023,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1], [2, 4, 5], [3]], 3)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2, 4, 5], [3]], 3)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0024","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f17c38fb1788d46062a8cb25aa750b0049017f609184ee679007397bac4ed759","family_id":"qf-mod30-quotient-classes","seed":202610304024,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0025","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75b5a194b8addae207c0ff9907e6b879c8707349e9acdc8c47496db3df20fd6e","family_id":"qf-mod30-quotient-classes","seed":202610304025,"semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"},"fields":{"title":"The Quotient Elements Are Entire Classes","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,count).","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], 2)","explanation":"Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], 2)."},"skills":["The Quotient Elements Are Entire Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0026","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85b87c4f4084f3322634ee4013ed8c48dab32b6ed0dbfbb95adc0b38cd1a098e","family_id":"qf-mod30-quotient-projection","seed":202610304026,"semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"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":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=0. Give (element,class).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0027","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edfbe26584106470ff0505d748454dec24a3229bdcac11bc58e7ad489a95e8b8","family_id":"qf-mod30-quotient-projection","seed":202610304027,"semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 2, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0028","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aacb3299fdf8f58542cfbc04bf44196e25b5fc84c62cec1c2d8f546abb4e7907","family_id":"qf-mod30-quotient-projection","seed":202610304028,"semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=2. Give (element,class).","answer":{"kind":"expression","value":"(2, [0, 2, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 2, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0029","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b04108b0e4a0dd383ec404b545a43b779c83662c8b286ae80341048e9d804a74","family_id":"qf-mod30-quotient-projection","seed":202610304029,"semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class).","answer":{"kind":"expression","value":"(3, [2, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [2, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0030","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca17cb6d2b71ebc8e38049c979c0ad4f54bcdd283c78018ecba0fa4188be1f77","family_id":"qf-mod30-quotient-projection","seed":202610304030,"semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=4. Give (element,class).","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [2, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0031","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5447eb5e66442979eca14b2912417c2cbbf549cfe77f036ba5d70a839443a41a","family_id":"qf-mod30-quotient-projection","seed":202610304031,"semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=5. Give (element,class).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 1, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0032","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483f7d9560077bdb32dc7d76f6487034c3ce4994298e0b1e1851691edd40bbc7","family_id":"qf-mod30-quotient-projection","seed":202610304032,"semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=0. Give (element,class).","answer":{"kind":"expression","value":"(0, [0, 2])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0033","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91a1c9bab8fb0f404ca272cbe999af3ba858473f6c58496fcee0b9b83da51844","family_id":"qf-mod30-quotient-projection","seed":202610304033,"semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).","answer":{"kind":"expression","value":"(1, [1, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0034","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ab6ef933b435dc8030b6405c57f7db88151ffa2d0fabeac1e31667582e43e25","family_id":"qf-mod30-quotient-projection","seed":202610304034,"semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=2. Give (element,class).","answer":{"kind":"expression","value":"(2, [1, 2])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [1, 2])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0035","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fc240b61bccd54f4ca2c8ff38f5317d1afd55c5cf9e64abf8e980c91d797fc8","family_id":"qf-mod30-quotient-projection","seed":202610304035,"semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"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":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class).","answer":{"kind":"expression","value":"(3, [3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0036","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7368531ae8c82227a6673c03771a3d768ce5f0ccef7c239b0dbb8c76af20c0ec","family_id":"qf-mod30-quotient-projection","seed":202610304036,"semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=4. Give (element,class).","answer":{"kind":"expression","value":"(4, [3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0037","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b34916a1a949fa5e8559ebf865a9c4619aca4923ebde7b751dec28103c1c359a","family_id":"qf-mod30-quotient-projection","seed":202610304037,"semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=5. Give (element,class).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 3, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0038","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3769c9de59f95cafc30a1dead694cba69ee818cae987995b783f63625964dd0","family_id":"qf-mod30-quotient-projection","seed":202610304038,"semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=0. Give (element,class).","answer":{"kind":"expression","value":"(0, [0, 2])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0039","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74208bff01fbe123b4d21997124051ff211fe632f7542d74938d56d77f2809f6","family_id":"qf-mod30-quotient-projection","seed":202610304039,"semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).","answer":{"kind":"expression","value":"(1, [0, 1])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [0, 1])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0040","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69cbe29ffdff420bc8b32b2443713b284a18dc1a524aeef9c36da96d319b1590","family_id":"qf-mod30-quotient-projection","seed":202610304040,"semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=2. Give (element,class).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 1, 2, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0041","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"519cfb024febb07aa63233d67890864211c44e9dd74311af5d67f95f061c0882","family_id":"qf-mod30-quotient-projection","seed":202610304041,"semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class).","answer":{"kind":"expression","value":"(3, [0, 3])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [0, 3])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0042","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccf9aded1796148d6343041793b99351b20a11cf66fe108266a8928c45aa4b96","family_id":"qf-mod30-quotient-projection","seed":202610304042,"semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=4. Give (element,class).","answer":{"kind":"expression","value":"(4, [1, 2, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [1, 2, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0043","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904aea5f6d0ab592ec0bc5f914cd42570839dcedfe40d177203d7098cf80434c","family_id":"qf-mod30-quotient-projection","seed":202610304043,"semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=5. Give (element,class).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 2, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0044","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739c5900bfc57a0dae17d8b5db97b8b23a1ad97bbd3438260d2b0cdb2135cb58","family_id":"qf-mod30-quotient-projection","seed":202610304044,"semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=0. Give (element,class).","answer":{"kind":"expression","value":"(0, [0, 1, 3])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 1, 3])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0045","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a63a006bc9d67a3dfd38f13d40327138703a157680ec78f179ed92edfac31a6d","family_id":"qf-mod30-quotient-projection","seed":202610304045,"semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=1. Give (element,class).","answer":{"kind":"expression","value":"(1, [1, 3])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 3])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0046","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3492ac71018996d7f5bdb6a36cab99578b20993c6cc118a9c0c97146d01c9f5c","family_id":"qf-mod30-quotient-projection","seed":202610304046,"semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=2. Give (element,class).","answer":{"kind":"expression","value":"(2, [0, 2, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 2, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0047","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"388c466e54fe0d4f7e73de0c2be4c8cf31dbd36143187815d084602b878237c8","family_id":"qf-mod30-quotient-projection","seed":202610304047,"semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class).","answer":{"kind":"expression","value":"(3, [0, 3])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [0, 3])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0048","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abf12cf56b3f6e0d021c7851afe525c7284d8e66b7c6e4ddbe5126d311112a28","family_id":"qf-mod30-quotient-projection","seed":202610304048,"semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=4. Give (element,class).","answer":{"kind":"expression","value":"(4, [2, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [2, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0049","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce936a4cb409be7676f106aded659ef217c324b8d4d1ed17bb34b944d6265930","family_id":"qf-mod30-quotient-projection","seed":202610304049,"semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=5. Give (element,class).","answer":{"kind":"expression","value":"(5, [0, 1, 4, 5])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 1, 4, 5])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0050","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"046ed45b79978d209441a55d049dcc31c89885ea9e48fe8b311053961d906a92","family_id":"qf-mod30-quotient-projection","seed":202610304050,"semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Canonical Projection Returns a Class, Not a Chosen Label","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=0. Give (element,class).","answer":{"kind":"expression","value":"(0, [0, 2, 3])","explanation":"Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2, 3])."},"skills":["Canonical Projection Returns a Class, Not a Chosen Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0051","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22b6ff59bc6a189db8a1ce23623ed26c7ec1c525ba17a6807ef1e6834f03a03a","family_id":"qf-mod30-quotient-partition","seed":202610304051,"semantic_template":"qf-mod30-quotient-partition/{\"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":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0052","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aa4a3cffdce66daf114e83bf8d437295448bbdc952000b044dbd32032850688","family_id":"qf-mod30-quotient-partition","seed":202610304052,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1, 2, 3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2, 3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0053","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39dfc931e5a30e463fb547a1028baa46574b37f6a95979d1941bad61fca33ce6","family_id":"qf-mod30-quotient-partition","seed":202610304053,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2, 3, 4, 5], [1]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 3, 4, 5], [1]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0054","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ba02a0bfbc9ad07ed1d2e203102fa8afd7544817ff2e2c6fa6c41285236f81f","family_id":"qf-mod30-quotient-partition","seed":202610304054,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0055","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9478e6bc0e92ef34b52be21f830f580a7e405268cf2061f8a511c70435bc6ca3","family_id":"qf-mod30-quotient-partition","seed":202610304055,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1], [2, 3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1], [2, 3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0056","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"291c308a3c7ac3fe4380cb74b822047cff29f24b7f476d7eb2eed4813b602f71","family_id":"qf-mod30-quotient-partition","seed":202610304056,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 3, 4, 5], [2]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 3, 4, 5], [2]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0057","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e412d180eb814113cbc8225f274674128bdf70bb365b16c7edc14128f3f3d46","family_id":"qf-mod30-quotient-partition","seed":202610304057,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0058","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cddd61b5176fa6647f7700a9923dbaa5a924e38d25a1a97c0782baec7c1555b","family_id":"qf-mod30-quotient-partition","seed":202610304058,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1, 3, 4, 5], [2]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 3, 4, 5], [2]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0059","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b88e39357c6feb8928bf1d23dc02da84a29babee4af6aa72f7e80ff1598ace22","family_id":"qf-mod30-quotient-partition","seed":202610304059,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1, 2]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1, 2]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0060","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"025d953e265240261bd26006d1ff23811f526bd20f0eb25d107a9c25c07179e7","family_id":"qf-mod30-quotient-partition","seed":202610304060,"semantic_template":"qf-mod30-quotient-partition/{\"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":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0061","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b76e4cb2d6e601845e60aa3414b1abd445e2fd0483b8d65e1969512b2a77546b","family_id":"qf-mod30-quotient-partition","seed":202610304061,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1, 2], [3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2], [3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0062","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"792277b5c23f3681db1b3b87ebb346ca4a36b269f8de028ab20b7c59f2ac573d","family_id":"qf-mod30-quotient-partition","seed":202610304062,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1], [2]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1], [2]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0063","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f563ca8fd78f043fc6019d52b5f824ff24bf1ba522e3f75a97b16c531a304fc","family_id":"qf-mod30-quotient-partition","seed":202610304063,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2], [1], [3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2], [1], [3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0064","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8670a1babb0da67dd21c07dc555a5f201527902edff87b23e134f0fb4ddfc6e","family_id":"qf-mod30-quotient-partition","seed":202610304064,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1], [2], [3, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2], [3, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0065","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80aa43290f306167846edc7c23b54fb62a7c90039025864a363c64a43b944d37","family_id":"qf-mod30-quotient-partition","seed":202610304065,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 2, 4, 5], [3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2, 4, 5], [3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0066","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82eb5a1840d7d88e7fda21a9503c673ac9bdbbe06943f0c6ba8880183b9412e5","family_id":"qf-mod30-quotient-partition","seed":202610304066,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0067","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4def4f42dcb1fc61a68345f6b22b13bb47af368ac4593d5b5b2211601e256276","family_id":"qf-mod30-quotient-partition","seed":202610304067,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1, 2, 4, 5], [3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2, 4, 5], [3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0068","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02e92ad2e75416359c08927b224b1b4ba25e2bc94875e7c95fa8c85be8309d5e","family_id":"qf-mod30-quotient-partition","seed":202610304068,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1, 3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1, 3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0069","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7931c5fac709093b433f426aeff1259add30a33e8837b130187160d6d32b045","family_id":"qf-mod30-quotient-partition","seed":202610304069,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0070","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52c5ea988e580e45acfddd952fd4ab9bcfd5a97eaefc28d98b3a5f2528d1fa3a","family_id":"qf-mod30-quotient-partition","seed":202610304070,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0], [1, 3], [2, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 3], [2, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0071","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"120dd9d59bcf4b32afea2a3a6b888d2608c15926452fd93ab39c785cecb180e3","family_id":"qf-mod30-quotient-partition","seed":202610304071,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1], [3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1], [3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0072","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cb5185f89972d4ec84d651e28fdc099b85af5f9bcd27943d5bb9a0d946d9027","family_id":"qf-mod30-quotient-partition","seed":202610304072,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 3], [1], [2, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3], [1], [2, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0073","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9e5b653a23e1cffe095eed72356d708f31611b244f91ece4d0eed76ea079e41","family_id":"qf-mod30-quotient-partition","seed":202610304073,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1], [2, 4, 5], [3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2, 4, 5], [3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0074","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72943063423c40004d9b1bebbd5f29c85fef461ab0242a43161215814b0a2fdc","family_id":"qf-mod30-quotient-partition","seed":202610304074,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0075","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b35415efe86131d3cfc929c46e2165256a4482065b5fea46fac4726ae5109f23","family_id":"qf-mod30-quotient-partition","seed":202610304075,"semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Verify Coverage and Disjointness of the Full Class Collection","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. Give (classes,covers?,disjoint?).","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], Yes, Yes)","explanation":"Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], Yes, Yes)."},"skills":["Verify Coverage and Disjointness of the Full Class Collection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0076","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca34a8ec6d27e37bd92febad15099ada9a57a9f9f565bf8992f23fcc8bd6556","family_id":"qf-mod30-quotient-representative","seed":202610304076,"semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"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":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0077","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646fe4ba27daf1f10da073af8058aa0a9707465d90d3d005ec0ecc6d9e75bdee","family_id":"qf-mod30-quotient-representative","seed":202610304077,"semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0078","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5739e3401ef22eda221914fff3519c1d67453a19ebeb4060530c990dc1e7cc1","family_id":"qf-mod30-quotient-representative","seed":202610304078,"semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(2, 1, No, [0, 2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0079","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d9992357a16421ea6bf7d04e89f32bf122391ffdf6011767412f9708666d9c6","family_id":"qf-mod30-quotient-representative","seed":202610304079,"semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0080","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381d10384a62cebe953e3c3f2f457f087c1de57c35f4de2a47e3700aa614a5d7","family_id":"qf-mod30-quotient-representative","seed":202610304080,"semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(4, 1, No, [2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0081","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe12793d313b76c6f2db0911de656b52fcce1b749b0bce7b05a7cecd5f713db","family_id":"qf-mod30-quotient-representative","seed":202610304081,"semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0082","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4456759aec5099de201210168011c52a70c594e963afee67241b9a2361859013","family_id":"qf-mod30-quotient-representative","seed":202610304082,"semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2], [1, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0083","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973bb768acaf9ef83895a0042beefb58606aa467fa8594ede411537d2ce0b891","family_id":"qf-mod30-quotient-representative","seed":202610304083,"semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0084","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"262a9a838d731ab5653199cfae9162049cb66bcd86560ef934412fa9fbefbb87","family_id":"qf-mod30-quotient-representative","seed":202610304084,"semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(2, 1, Yes, [1, 2], [1, 2], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, Yes, [1, 2], [1, 2], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0085","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2ff27572be1ff845a2593cd74e9d216208373e0059297222008e596badaf23d","family_id":"qf-mod30-quotient-representative","seed":202610304085,"semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"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":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0086","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02bd6ef75533c416d7d0b4388bcb712d5eacf241ce3a316d94b3a111d35e6f6","family_id":"qf-mod30-quotient-representative","seed":202610304086,"semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(4, 1, No, [3, 4, 5], [1, 2], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [3, 4, 5], [1, 2], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0087","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66033b403fb038a3029f5c6acb66809c2c21636969f227dcddb3a1cfd02a90ab","family_id":"qf-mod30-quotient-representative","seed":202610304087,"semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0088","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"847a19eb896e099bf25c63f6674f6fb6d2593dc4431b3cf298f30baca5f727ef","family_id":"qf-mod30-quotient-representative","seed":202610304088,"semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2], [1], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0089","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8af6b915c9055fd55712894e76e7ddea3bbbd99056f5e983775b917d8b425d","family_id":"qf-mod30-quotient-representative","seed":202610304089,"semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(1, 4, No, [0, 1], [3, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, No, [0, 1], [3, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0090","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77aa6372ae2e863ebef4562dee0b1369512eb66a76570966c7eab44bd6be5825","family_id":"qf-mod30-quotient-representative","seed":202610304090,"semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0091","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb2eaff48e04846d177f4cc6b217fa33e6f732886176d0a230dd170a633e8c29","family_id":"qf-mod30-quotient-representative","seed":202610304091,"semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(3, 4, No, [0, 3], [1, 2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [1, 2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0092","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a712cc4c08f00119949ec0c8e978f771399d71ef49dd79150af721622a7539","family_id":"qf-mod30-quotient-representative","seed":202610304092,"semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0093","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04a142c28b2fddca109d696a0caf4ca16db0eab2dc3eae568a1464e33f55b45d","family_id":"qf-mod30-quotient-representative","seed":202610304093,"semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0094","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7bb549aa8c76285d46e7ba5c26380f423fafdd05efdfbb8dc4e96db2905c11e","family_id":"qf-mod30-quotient-representative","seed":202610304094,"semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0095","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5542815552ed02741173c752911849b9a01ce81614bb63a31c74dabecf0b39eb","family_id":"qf-mod30-quotient-representative","seed":202610304095,"semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(1, 4, No, [1, 3], [2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, No, [1, 3], [2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0096","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"451c89fd3de58d6135a2ec278e569de833dd0df68d5d8631a6ef5dac0bcb4e64","family_id":"qf-mod30-quotient-representative","seed":202610304096,"semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(2, 1, No, [0, 2, 4, 5], [1], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 4, 5], [1], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0097","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"838dcd7e338826b73fb5361a02d921579f972a43e5c74757013f24695782a36a","family_id":"qf-mod30-quotient-representative","seed":202610304097,"semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(3, 4, No, [0, 3], [2, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [2, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0098","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1def7afc9b38cc7e1d9d47593ffd3d3a7e792bdeb85c3e77dbe39d77b779228","family_id":"qf-mod30-quotient-representative","seed":202610304098,"semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(4, 1, No, [2, 4, 5], [0, 1], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [2, 4, 5], [0, 1], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0099","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b0cdd03c79efce9d4240d3b3e1a2728be51a4c32b69064f93e6f8c777b05ed3","family_id":"qf-mod30-quotient-representative","seed":202610304099,"semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r04-quotient-set-0100","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c8160821a94b3823c3b7eb5bc5155cd12301c749d2d82a0db23edbc6434b3fc","family_id":"qf-mod30-quotient-representative","seed":202610304100,"semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"},"fields":{"title":"Representative Invariance Does Not Define Quotient Operations","stem":"Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2, 3], [1, 4, 5], Yes, no_quotient_operation_claim)","explanation":"Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2, 3], [1, 4, 5], Yes, no_quotient_operation_claim)."},"skills":["Representative Invariance Does Not Define Quotient Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}