{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0001","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"804b3ea4efed597ca4e0e354758e412de104a30df7d356e0614b54c1db42e60b","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113001,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a0, y=b0. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a0, b0, [[a0, c0], [b0]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a0, b0, [[a0, c0], [b0]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0002","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e13ff0573a3aadc57324cdc8601680ac2a7bcac9dea80ac9519a22209296745","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113002,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a1, y=b1. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a1, b1, [[a1, c1], [b1]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a1, b1, [[a1, c1], [b1]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0003","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd20ab231ebf117e44c801d8b59078d66f260ac1a98df46a80e7388c44a3992e","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113003,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a2, y=b2. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a2, b2, [[a2, c2], [b2]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a2, b2, [[a2, c2], [b2]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0004","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"107b9cfe56396b4b3a3c8a0571b7e0d7db693a0f45eef57f2878789c0f702ee4","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113004,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a3, y=b3. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a3, b3, [[a3, c3], [b3]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a3, b3, [[a3, c3], [b3]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0005","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d1b4f57c1c7a5b943be13a138bfbd9e560dfac8774f5134d3d6ef8dfd632ef2","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113005,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a4, y=b4. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a4, b4, [[a4, c4], [b4]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a4, b4, [[a4, c4], [b4]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0006","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df42b0d973718696f0a6c5dceeac59027938176335144ba95ee3d4a811c6cd31","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113006,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a5, y=b5. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a5, b5, [[a5, c5], [b5]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a5, b5, [[a5, c5], [b5]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0007","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d461525dd60e03b93fa872d63e1a4c8b2186693a9c249610eae9df76294f212f","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113007,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a6, y=b6. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a6, b6, [[a6, c6], [b6]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a6, b6, [[a6, c6], [b6]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0008","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040a357b69c5bd6f8c1cef6fbe648654cc372174cff177e835d14d800c7b46cc","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113008,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a7, y=b7. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a7, b7, [[a7, c7], [b7]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a7, b7, [[a7, c7], [b7]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0009","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897c09a005df3165d1f0980fefc6394d24427dc52f06ef908490a56aa26d333c","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113009,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a8, y=b8. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a8, b8, [[a8, c8], [b8]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a8, b8, [[a8, c8], [b8]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0010","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72ce00cb2ff3a41ed262769d87cec2d341af76559197038bb0402648d135fd32","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113010,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a9, y=b9. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a9, b9, [[a9, c9], [b9]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a9, b9, [[a9, c9], [b9]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0011","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82a09c4a3457422b6e59a19ec66bd9ca3dbbb70f45a60bece53bc386fbd25b50","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113011,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a10, y=b10. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a10, b10, [[a10, c10], [b10]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a10, b10, [[a10, c10], [b10]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0012","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"410a70a16ae4112030ff24c7499025ddaad07418d57bb70510f7a591d3500a3c","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113012,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a11, y=b11. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a11, b11, [[a11, c11], [b11]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a11, b11, [[a11, c11], [b11]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0013","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"010b63a0dd1ce988ba24e862ed9f81c6775eea2b4e51b293f11159bc270177d3","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113013,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a12, y=b12. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a12, b12, [[a12, c12], [b12]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a12, b12, [[a12, c12], [b12]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0014","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c501f88fbb2519a063e1753dd6b9ba9e35071633a522602354b0ae789c7a36d9","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113014,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a13, y=b13. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a13, b13, [[a13, c13], [b13]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a13, b13, [[a13, c13], [b13]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0015","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2acca7f1a6dbbcdf367170e8af290e229fad56efad01e2ae1a1104ccae442569","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113015,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a14, y=b14. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a14, b14, [[a14, c14], [b14]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a14, b14, [[a14, c14], [b14]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0016","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98ffa810c49828749c1d528f4598f198fe8b1fadd55f009bd2b71cecddf7efc5","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113016,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a15, y=b15. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a15, b15, [[a15, c15], [b15]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a15, b15, [[a15, c15], [b15]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0017","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaa1c709b6fcc15157bb3e8951e3ad4669c6f523c0e961ffd77037c3d6320bcd","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113017,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a16, y=b16. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a16, b16, [[a16, c16], [b16]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a16, b16, [[a16, c16], [b16]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0018","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f785ebcebf2ece3bb196418ae50611c381600a06ba224b84ab0e4829389ed262","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113018,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a17, y=b17. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a17, b17, [[a17, c17], [b17]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a17, b17, [[a17, c17], [b17]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0019","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66b0b1a670a6db982c6c1214499ef1554e4cfaebcc5b038bceb23446d1576f12","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113019,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a18, y=b18. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a18, b18, [[a18, c18], [b18]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a18, b18, [[a18, c18], [b18]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0020","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7edd3a06e73c860608bb6ddee03180175762191f9f018a98f134be3fc2e3f72a","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113020,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a19, y=b19. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a19, b19, [[a19, c19], [b19]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a19, b19, [[a19, c19], [b19]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0021","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"393785ef8528378d8330c26ab7a8f17c01b4c6b994167f1785e4240fec35d1e5","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113021,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a20, y=b20. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a20, b20, [[a20, c20], [b20]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a20, b20, [[a20, c20], [b20]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0022","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb35e81b1b26e675bb1d1271d2733d531de012acab7a3445f98e7443e7e69643","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113022,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a21, y=b21. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a21, b21, [[a21, c21], [b21]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a21, b21, [[a21, c21], [b21]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0023","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219f52f6834a0264a673a158d14964b5288ee1d63c4345f6082ac5b6fa7e00af","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113023,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a22, y=b22. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a22, b22, [[a22, c22], [b22]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a22, b22, [[a22, c22], [b22]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0024","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c49b6b6c587becb20af9173d1cb252f73570539e5d3180d224d77c86695d5302","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113024,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a23, y=b23. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a23, b23, [[a23, c23], [b23]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a23, b23, [[a23, c23], [b23]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0025","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"401468148180919f96a573614d913994a7ccb1d14294acd6c26dba0023477a71","family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113025,"semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}"},"fields":{"title":"A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods","stem":"Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a24, y=b24. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).","answer":{"kind":"expression","value":"(a24, b24, [[a24, c24], [b24]], Yes, Yes)","explanation":"Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a24, b24, [[a24, c24], [b24]], Yes, Yes)."},"skills":["A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0026","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"838d83883925cf04be061de9fa1d0453e2bbd23cb70da1aff150a4379f9d25fe","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113026,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0027","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eb5efbb83df3442b3d782dbc5ed33f24934a5fc08bfd40abba58f5845ab392f","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113027,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0028","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06586aab07b2367ce236f75bf890f7eab53c7bbfd7286d3f3ace3c0430119fa7","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113028,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0029","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"574188e746fa95b311e1348d659894a2b62df2e60746f722e301f3aef81cdf2b","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113029,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0030","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bbbbafbcac897db7476bb7e2e6279201d1429517faa8d637c730888adaa1b6b","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113030,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0031","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43abb26c4d8b68e1da69bfa78255faef1f0a9bc623b2a44236f30546d43b8cf4","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113031,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0032","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa2a322f23e3cb429dafb6bb0eb967c7d2ba95350f1fc6c99de7ae04b9398117","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113032,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0033","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d54011b30482f8b871637fe0da5469061b840e643864cc239c091198ca059f2d","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113033,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0034","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4135c96cb88ab023176f0a6de85a52e6cd8b41ef911e58740445057d6c646fe","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113034,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0035","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0327c8b375920a870408d8d367f2dc9f6db4afc1f682fab8f8523d89a611e6e9","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113035,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0036","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d50eb829c61fd7aacc6263f8d4fa984bb8f0935169f820ad7616f0f9eb61cb41","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113036,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0037","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"654987fc41bdff1f1dc868d944716d6d2a9931a10b16bae2dd8417b329d51189","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113037,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0038","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd7725a5e11f0e3ab8b44a8d0f1d113f1801fb1104619ffeb19a89249307e8df","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113038,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0039","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8375af2f6c8e32c6e3786087f8d5ca2bac25b78b03f5ad2e41c2de42f09dbb7","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113039,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0040","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d75a4a9760cb371c48bdffd386f6433db9a89052736e0a4367f44896a29da01b","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113040,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0041","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f963bfe9fab1bf7335ac155d0f5389fa8dbcd69a85d963eea231216fc31fe14c","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113041,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0042","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c19936d28b812031293f4f29134bbe31695b0186b1736de35f89d2455cd825","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113042,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0043","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"530dfbb9ef9748f527c4e1c1bb1a81b061dcf9e00a128219a71fc15ccaa55f09","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113043,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0044","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90098ced916be2dd6c53367a1b5601a66b3765fda2acda8a431261c0e27cb9fc","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113044,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0045","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d91bb23b9926c9512cfbbd16cd2d23ccd84586082ee4e3f5b353d2a60509daa","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113045,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0046","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c628409a9a487bd476cea557a3bcf9a2d4dcfddc49badf15dd7bcc102b95abb","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113046,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0047","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68e92af8d23f1070bdde8720b8ff5966a5352a7f4eb87ffd3de7683cc8304f25","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113047,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0048","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b57e90552c9757bad6a61ef8d5ec0aeaccd1ea0298eb63e0435f9e964632a9","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113048,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0049","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6437802853bb949dd74dd0a94bdcfd64309a27e5341f1d6ff928241aa629536b","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113049,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0050","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb35260d79de230cb4ce52571f1c8b7bef32ea4c13162e9f21ed8015583579cc","family_id":"qf-mod42-hausdorff-full-audit","seed":202611113050,"semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}"},"fields":{"title":"All Distinct Point Pairs Must Be Separated in the Complete Finite Topology","stem":"Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).","answer":{"kind":"expression","value":"([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, all_distinct_pairs_exhaustively_separated)","explanation":"Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, all_distinct_pairs_exhaustively_separated)."},"skills":["All Distinct Point Pairs Must Be Separated in the Complete Finite Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0051","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b37c6c6240dd5919300a336b2cfd8119c48a20cc59776b6cc588792ce3fdbaea","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113051,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a0, y=b0. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a0, b0, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a0, b0, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0052","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf44ae12349f12dedab45c8fb54cfa1cd50e569c4010c8116bdeb822e18be0d","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113052,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a1, y=b1. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a1, b1, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a1, b1, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0053","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee5b218c52ee554c9d90deeece68dd217e2e9079e7173c35acc919533505c2d6","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113053,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a2, y=b2. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a2, b2, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a2, b2, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0054","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef5fd593640d2189c00f8ce6d5a86dc0027efeda835406c6549d4488d5f709f1","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113054,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a3, y=b3. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a3, b3, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a3, b3, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0055","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9496ab7f8714f9238753528ed7133573520043bdaa19a4160b0acedbe345451","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113055,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a4, y=b4. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a4, b4, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a4, b4, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0056","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"097c2a57a39456cecf9f4c8dd0d9685572936acf30dfa8d3ea83544994e7b8fe","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113056,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a5, y=b5. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a5, b5, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a5, b5, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0057","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a81b8a37cae724e2b917cb4d59e0b15b49e4cd7e88153015110b84dca3166377","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113057,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a6, y=b6. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a6, b6, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a6, b6, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0058","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832c45fc0a84d630202bf9b3b0bb207e6ae778edb80dc190f065113be5caf3af","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113058,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a7, y=b7. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a7, b7, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a7, b7, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0059","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"223593825983f175e93dac667f4d8c10929b9f951c4fc0f990531dd988c107cf","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113059,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a8, y=b8. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a8, b8, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a8, b8, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0060","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b9a92f5e8b61f308100a6bb867fe5ba65912c4381cd331c84a9db8dfecfa20","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113060,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a9, y=b9. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a9, b9, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a9, b9, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0061","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0693303bf3c159236edfd04229bda24ba9a1d452f08fc1a0c2cb99eb91944f20","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113061,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a10, y=b10. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a10, b10, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a10, b10, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0062","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e82654a0c282748766447de24cdcabcc38abfd831c0c7459d04a78829e04774c","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113062,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a11, y=b11. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a11, b11, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a11, b11, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0063","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cd8aa68ec9f5a4d198c3db1cc62d62e996b2d5dd21834fdd392e6edb7bc8f88","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113063,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a12, y=b12. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a12, b12, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a12, b12, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0064","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"286bfefd50522636dc35243b332e42b72abfbc91eda1f014593d9130018a70ad","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113064,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a13, y=b13. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a13, b13, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a13, b13, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0065","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de359592ca41942df0bc80d74e805be6e41312e386c11fec0853680dec8577e0","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113065,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a14, y=b14. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a14, b14, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a14, b14, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0066","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ccd33c15029dd68aaeece2b92f8e9d6d80ba5f292bf6923872bebce5a8221f6","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113066,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a15, y=b15. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a15, b15, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a15, b15, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0067","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48bce6cd5b80284baad63ad1621cc68f2a8966f3660a009eb602c942e22823e1","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113067,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a16, y=b16. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a16, b16, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a16, b16, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0068","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37b0c9379c743e371f2e22e2defacc836e7bfce996b7d03f453c06083563e07c","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113068,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a17, y=b17. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a17, b17, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a17, b17, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0069","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4ffc71470ada83ece5ce049b18bd392993b54ae14640deec8ee3773fef7fff","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113069,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a18, y=b18. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a18, b18, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a18, b18, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0070","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95a27def4d4d4ac1624a2f25b8593cdaea59db1672bc169ea8c2452348d04f38","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113070,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a19, y=b19. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a19, b19, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a19, b19, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0071","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cf698163208782957397f9c63e91578c56c2dbdcc11cd4bf7b63a438b50cdd0","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113071,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a20, y=b20. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a20, b20, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a20, b20, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0072","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6596c86a3fb836394aa785edf5611246eb71e1987d48aa5f5b3b715cdd6f387f","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113072,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a21, y=b21. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a21, b21, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a21, b21, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0073","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3e933aff149fa51ab6a341a316489094979590fe001410533133fcb6be4d003","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113073,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a22, y=b22. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a22, b22, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a22, b22, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0074","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bacaea29cd4275697bfac7ddce994a0b7d9754b0b6746ecd946125cfaaac9794","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113074,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a23, y=b23. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a23, b23, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a23, b23, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0075","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edce287be3cd48f51154ad4cb4eb30421c268b65dfc82d8a2cde197c33b086b4","family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113075,"semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}"},"fields":{"title":"One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness","stem":"Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a24, y=b24. Give (x, y, witness_or_null, separated, boundary).","answer":{"kind":"expression","value":"(a24, b24, null, No, one_failed_distinct_pair_proves_non_Hausdorff)","explanation":"Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a24, b24, null, No, one_failed_distinct_pair_proves_non_Hausdorff)."},"skills":["One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0076","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811fdd77f2ea9fb46e707369d33648da48238d0d10d5c9349524a2e08497fec1","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113076,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a0\",\"b0\",\"c0\"],\"sampled_pairs\":[[\"c0\",\"a0\"]],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"c0\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c0, a0]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c0, a0, [[c0], [a0, b0]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c0, a0, [[c0], [a0, b0]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0077","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5938fea990f120de663a7da39c504bc590735ac0d7b162dc7982548b1304df9","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113077,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a1\",\"b1\",\"c1\"],\"sampled_pairs\":[[\"c1\",\"a1\"]],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"c1\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c1, a1]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c1, a1, [[c1], [a1, b1]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c1, a1, [[c1], [a1, b1]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0078","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b26bd06a40a271ed7c62a5d1600c3e45a64c6190ab35ea114e79f709dcf27724","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113078,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a2\",\"b2\",\"c2\"],\"sampled_pairs\":[[\"c2\",\"a2\"]],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"c2\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c2, a2]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c2, a2, [[c2], [a2, b2]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c2, a2, [[c2], [a2, b2]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0079","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce1f000c4be37ca166505b1d277e192b44d860e8efb85d5ace7df58f5de37f48","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113079,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a3\",\"b3\",\"c3\"],\"sampled_pairs\":[[\"c3\",\"a3\"]],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"c3\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c3, a3]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c3, a3, [[c3], [a3, b3]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c3, a3, [[c3], [a3, b3]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0080","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad6cf01aa8acb61960fa5482daab0ed6dc6887352b75fc279c7e83f4251918d0","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113080,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a4\",\"b4\",\"c4\"],\"sampled_pairs\":[[\"c4\",\"a4\"]],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"c4\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c4, a4]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c4, a4, [[c4], [a4, b4]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c4, a4, [[c4], [a4, b4]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0081","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42fd5ed0c17e7cfe913eb381e1b66951981b5133c73caa84c7353a5e61a65baf","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113081,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a5\",\"b5\",\"c5\"],\"sampled_pairs\":[[\"c5\",\"a5\"]],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"c5\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c5, a5]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c5, a5, [[c5], [a5, b5]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c5, a5, [[c5], [a5, b5]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0082","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2128cf58bddb6c59087a9a1d93e8a493339c0bc9d2ab75e0bd3858b430e70b4e","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113082,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a6\",\"b6\",\"c6\"],\"sampled_pairs\":[[\"c6\",\"a6\"]],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"c6\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c6, a6]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c6, a6, [[c6], [a6, b6]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c6, a6, [[c6], [a6, b6]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0083","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e06f5ee6092c264be291282507636810c0a49fcc3ef6df9fcb87da8d8817bbd","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113083,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a7\",\"b7\",\"c7\"],\"sampled_pairs\":[[\"c7\",\"a7\"]],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"c7\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c7, a7]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c7, a7, [[c7], [a7, b7]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c7, a7, [[c7], [a7, b7]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0084","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17c23e7e8d5f2f6698e0f22465553064493582cff149049e1b7dcc0caed25734","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113084,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a8\",\"b8\",\"c8\"],\"sampled_pairs\":[[\"c8\",\"a8\"]],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"c8\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c8, a8]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c8, a8, [[c8], [a8, b8]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c8, a8, [[c8], [a8, b8]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0085","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c72f4e985f9996aa3bc1671cef3c9ecf2f40dc9e6819577393c15c4bb71a76e","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113085,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a9\",\"b9\",\"c9\"],\"sampled_pairs\":[[\"c9\",\"a9\"]],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"c9\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c9, a9]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c9, a9, [[c9], [a9, b9]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c9, a9, [[c9], [a9, b9]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0086","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81867bc730b451f0d828a0df546b65026ad0c085312832bd542e1e655b3a4296","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113086,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a10\",\"b10\",\"c10\"],\"sampled_pairs\":[[\"c10\",\"a10\"]],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"c10\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c10, a10]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c10, a10, [[c10], [a10, b10]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c10, a10, [[c10], [a10, b10]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0087","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b78a3bb2282e2f4cd6b364a97c690c770c08daaaf6dc3cd8df4045246ac9e74","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113087,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a11\",\"b11\",\"c11\"],\"sampled_pairs\":[[\"c11\",\"a11\"]],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"c11\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c11, a11]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c11, a11, [[c11], [a11, b11]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c11, a11, [[c11], [a11, b11]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0088","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da7bea9a8d9cd73bbb80b9aa8844c72f0843669ecd4ccbe8bf62a32a42171a4","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113088,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a12\",\"b12\",\"c12\"],\"sampled_pairs\":[[\"c12\",\"a12\"]],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"c12\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c12, a12]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c12, a12, [[c12], [a12, b12]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c12, a12, [[c12], [a12, b12]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0089","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"735d2c2c55c2b1174c9b1e0655e3bbfd114cc187e53462a77c87cb9d7c3b94b6","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113089,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a13\",\"b13\",\"c13\"],\"sampled_pairs\":[[\"c13\",\"a13\"]],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"c13\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c13, a13]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c13, a13, [[c13], [a13, b13]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c13, a13, [[c13], [a13, b13]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0090","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bffdd29d4a4ab5d2346a59c7ec0810e0428f15f1bf1cce7e407911636b79f5c","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113090,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a14\",\"b14\",\"c14\"],\"sampled_pairs\":[[\"c14\",\"a14\"]],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"c14\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c14, a14]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c14, a14, [[c14], [a14, b14]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c14, a14, [[c14], [a14, b14]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0091","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1c9db99e6e3bb5681042dbeb0538caadac6289b9562b9007e8c3c327efb2e83","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113091,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a15\",\"b15\",\"c15\"],\"sampled_pairs\":[[\"c15\",\"a15\"]],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"c15\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c15, a15]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c15, a15, [[c15], [a15, b15]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c15, a15, [[c15], [a15, b15]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0092","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7477ac924238bad9e796d52a6493223eb41b194adbc72821766cbc04509964a4","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113092,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a16\",\"b16\",\"c16\"],\"sampled_pairs\":[[\"c16\",\"a16\"]],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"c16\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c16, a16]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c16, a16, [[c16], [a16, b16]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c16, a16, [[c16], [a16, b16]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0093","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7e57247a1f8250bce20516aba04ca4c9e97a517ce11f31e78660e876d1ec81","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113093,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a17\",\"b17\",\"c17\"],\"sampled_pairs\":[[\"c17\",\"a17\"]],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"c17\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c17, a17]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c17, a17, [[c17], [a17, b17]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c17, a17, [[c17], [a17, b17]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0094","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6ec517d46bbeffa8287ce770c796fe73f3b2b78d60f22bcb7b2f6e891c5f685","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113094,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a18\",\"b18\",\"c18\"],\"sampled_pairs\":[[\"c18\",\"a18\"]],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"c18\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c18, a18]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c18, a18, [[c18], [a18, b18]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c18, a18, [[c18], [a18, b18]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0095","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19d5058f4446391916782f3cea9e9641530bcba6b87809d1199db26cd06008e0","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113095,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a19\",\"b19\",\"c19\"],\"sampled_pairs\":[[\"c19\",\"a19\"]],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"c19\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c19, a19]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c19, a19, [[c19], [a19, b19]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c19, a19, [[c19], [a19, b19]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0096","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d483103ec43fce0c6d0ae701b9002bd30e0f0b053265654588133656c783c41","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113096,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a20\",\"b20\",\"c20\"],\"sampled_pairs\":[[\"c20\",\"a20\"]],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"c20\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c20, a20]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c20, a20, [[c20], [a20, b20]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c20, a20, [[c20], [a20, b20]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0097","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e434b3aaa44881c6b299d0254d51f824954ad2ae258bc08711727cee925f2a01","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113097,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a21\",\"b21\",\"c21\"],\"sampled_pairs\":[[\"c21\",\"a21\"]],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"c21\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c21, a21]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c21, a21, [[c21], [a21, b21]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c21, a21, [[c21], [a21, b21]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0098","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45f41bd0ffa8833618c12c493bac48bcbfd38e3cdda552358a1232172c93fc85","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113098,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a22\",\"b22\",\"c22\"],\"sampled_pairs\":[[\"c22\",\"a22\"]],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"c22\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c22, a22]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c22, a22, [[c22], [a22, b22]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c22, a22, [[c22], [a22, b22]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0099","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bf5c126bdf3a840453440412f2be67b5142ffde5798c395234c75e70550e3f7","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113099,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a23\",\"b23\",\"c23\"],\"sampled_pairs\":[[\"c23\",\"a23\"]],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"c23\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c23, a23]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c23, a23, [[c23], [a23, b23]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c23, a23, [[c23], [a23, b23]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r03-hausdorff-space-0100","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cddf77c9e32161a316bf97d4b339ef838671c02f26b5505c2a31e322fef6a653","family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113100,"semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a24\",\"b24\",\"c24\"],\"sampled_pairs\":[[\"c24\",\"a24\"]],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"c24\"],[]]}"},"fields":{"title":"Successful Pair Samples Do Not Certify Unchecked or Infinite Cases","stem":"Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c24, a24]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).","answer":{"kind":"expression","value":"([[c24, a24, [[c24], [a24, b24]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation":"Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c24, a24, [[c24], [a24, b24]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)."},"skills":["Successful Pair Samples Do Not Certify Unchecked or Infinite Cases"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}