{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0001","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f39d8bd458f3c6348efd544e6f838f9c80ad4fd71301e8a73bda075abef2b5f","family_id":"qf-mod26-joint-qualification","seed":202610264001,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 3/8] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -1], [-11, 0], [-10, 1], [-9, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0002","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811c296f890d23a5b8692598c6f8921257ead529cdd1cb4ef34e8883f374497a","family_id":"qf-mod26-joint-qualification","seed":202610264002,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -1], [-10, 0], [-9, 1], [-8, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0003","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d8be6f4190faf1a252e2e0888eaa2c41d6383d20f2817c28d0123a0b5f48e42","family_id":"qf-mod26-joint-qualification","seed":202610264003,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-10, -1], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1], [-9, 0], [-8, 1], [-7, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0004","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50f1f76828891bab4be87cbda6bd4a9d119a3f1a00dbcec86355bc3a9dfaf2ee","family_id":"qf-mod26-joint-qualification","seed":202610264004,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, -1], [-8, 0], [-7, 1], [-6, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0005","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30cd0cb6a772f28dd1ab233ec8b0c1adbc12775200c706ffa123be76a3b7eb7a","family_id":"qf-mod26-joint-qualification","seed":202610264005,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, -1], [-7, 0], [-6, 1], [-5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0006","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"874bffd8c026647b1f498222f681b79777f075b90687a0d6c4350003b554034d","family_id":"qf-mod26-joint-qualification","seed":202610264006,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 3/8] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, -1], [-6, 0], [-5, 1], [-4, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0007","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b682e12dcfb8a6abb5bac3107cfe3b6d7c6880a5f444db37a410e80c6f09cd0a","family_id":"qf-mod26-joint-qualification","seed":202610264007,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, -1], [-5, 0], [-4, 1], [-3, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0008","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a69d638af0f5079dfa3f39adfd92d95748b24d63371cbc76a9f9b0bea09eee88","family_id":"qf-mod26-joint-qualification","seed":202610264008,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -1], [-4, 0], [-3, 1], [-2, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0009","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"415d5147802011fa42a9ac2d1f5ce795ee705a738af4004868f092668288dae3","family_id":"qf-mod26-joint-qualification","seed":202610264009,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -1], [-3, 0], [-2, 1], [-1, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0010","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d16be79d68689ad705271b85c7ffe75e092a306a95158b81e0594c0093584d5","family_id":"qf-mod26-joint-qualification","seed":202610264010,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-3, -1], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1], [-2, 0], [-1, 1], [0, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0011","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f34bd543e357e9cf26ab4ce9705a1df456d5c67bac9df1e159af7be69b56e6a","family_id":"qf-mod26-joint-qualification","seed":202610264011,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 3/8] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, -1], [-1, 0], [0, 1], [1, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0012","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a10d6cd9fb7632e4d09274784bb03ab544e37ee50cebe3f09acc15f8821f64b8","family_id":"qf-mod26-joint-qualification","seed":202610264012,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[-1, -1], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, -1], [0, 0], [1, 1], [2, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0013","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf13e738b62f797c3e36f296323e555a45d5044755d560c586c235a599b776eb","family_id":"qf-mod26-joint-qualification","seed":202610264013,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[0, -1], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, -1], [1, 0], [2, 1], [3, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0014","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bcc5a5726674a165a51807936f99111e049734893c058744f8cf2ac251ac7dd","family_id":"qf-mod26-joint-qualification","seed":202610264014,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[1, -1], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, -1], [2, 0], [3, 1], [4, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0015","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381026df088af63e6f8f3e54c5e3d2b75c093a130429ac67c445e01d8fc0ce14","family_id":"qf-mod26-joint-qualification","seed":202610264015,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [3, 0], [4, 1], [5, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0016","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c6425d7cf270872875f74cc8e3f317365d41733872ad2e695de64ce1cab6ee9","family_id":"qf-mod26-joint-qualification","seed":202610264016,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 3/8] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[3, -1], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -1], [4, 0], [5, 1], [6, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0017","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84d9878e9af71e47d8b1cee547566577b138110905eb0a7b01be43adbe7d81f6","family_id":"qf-mod26-joint-qualification","seed":202610264017,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [5, 0], [6, 1], [7, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0018","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c633f18adc34bd39051eb787427aa053b183e4519615a462f1642d11c7953b31","family_id":"qf-mod26-joint-qualification","seed":202610264018,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[5, -1], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, -1], [6, 0], [7, 1], [8, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0019","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ca5d642a3039c1c1de86e8383823642aa187156100de324095339d5a1b15ec","family_id":"qf-mod26-joint-qualification","seed":202610264019,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [7, 0], [8, 1], [9, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0020","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c54ff580f75d68e9e108dab5839890256a4b501d34575fb12ea4c2ee44fe298e","family_id":"qf-mod26-joint-qualification","seed":202610264020,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, -1], [8, 0], [9, 1], [10, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0021","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09252fdce24485306669d38b4320b133edcfa11f876b17bc832787c2cb91ee67","family_id":"qf-mod26-joint-qualification","seed":202610264021,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[3,8]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 3/8] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, -1], [9, 0], [10, 1], [11, 2]], [1/8, 1/4, 1/8, 3/8], Yes, 7/8, No)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0022","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc0c2b9f8e614f1c98e9df2aacbf4a59e07721a497f233b5ec99c9a67b4ec0f","family_id":"qf-mod26-joint-qualification","seed":202610264022,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -1], [10, 0], [11, 1], [12, 3]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0023","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4307cd3ab5f60c7f5b51713a12da3798082fa8217114bb52eecc3e82c558c2d","family_id":"qf-mod26-joint-qualification","seed":202610264023,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[10, -1], [11, 0], [12, 1], [13, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[10, -1], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [11, 0], [12, 1], [13, 4]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0024","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e80e377ce051e92f76bded3f10c7a61abd511b5086f7ca106630aa5acf2add29","family_id":"qf-mod26-joint-qualification","seed":202610264024,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [12, 0], [13, 1], [14, 5]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0025","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e6e17f3ddecf3089b2655c88f6537329e0e739ef01d51f91532a7d94a83ed86","family_id":"qf-mod26-joint-qualification","seed":202610264025,"semantic_template":"qf-mod26-joint-qualification/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Qualify a Whole Exact Joint Probability Table","stem":"Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (support,normalized masses,all masses nonnegative?,exact sum,qualified?).","answer":{"kind":"expression","value":"([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)","explanation":"Keep the original pair order and aligned normalized masses. Check nonnegativity and exact total separately; a nearly normalized or signed table is not qualified. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, -1], [13, 0], [14, 1], [15, 6]], [1/8, 1/4, 1/8, 1/2], Yes, 1, Yes)."},"skills":["Qualify a Whole Exact Joint Probability Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0026","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81fc984e9a73b9d11b6f5bed6c2b8e6e83183f3ed14fb66167ae09104eb0af81","family_id":"qf-mod26-joint-marginals","seed":202610264026,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, 1/8], [-11, 1/4], [-10, 1/8], [-9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0027","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"027883c83437c63ad49301d6dac11d65f94d88c4e19122b2f5076071aa967714","family_id":"qf-mod26-joint-marginals","seed":202610264027,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, 1/8], [-10, 1/4], [-9, 1/8], [-8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0028","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53bc6ad6a625823933eb23168f7d3e57ff76d048f22b14afc2f26a386c0ae461","family_id":"qf-mod26-joint-marginals","seed":202610264028,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, 1/8], [-9, 1/4], [-8, 1/8], [-7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0029","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef87beeedd6334380a3e3feb125c3f6ae127907d66cd2d5b81f2f3ee41499f74","family_id":"qf-mod26-joint-marginals","seed":202610264029,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 1/8], [-8, 1/4], [-7, 1/8], [-6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0030","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0fd445bac301e6e1d78a707d3156dc9e02509ec62e683180007263091390be0","family_id":"qf-mod26-joint-marginals","seed":202610264030,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1/8], [-7, 1/4], [-6, 1/8], [-5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0031","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a7c742b052b00ed371406b5510a3ee77bee31b1e8180838822bdc0bb5f29137","family_id":"qf-mod26-joint-marginals","seed":202610264031,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 1/8], [-6, 1/4], [-5, 1/8], [-4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0032","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c1746ece051dc5a1c63f8500304e20d524e4ed83b484bd2fb5d645ca00cf89a","family_id":"qf-mod26-joint-marginals","seed":202610264032,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 1/8], [-5, 1/4], [-4, 1/8], [-3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0033","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbad75f06f6468e798ed64421df35e41066c06a8e816958153223655f30d6a71","family_id":"qf-mod26-joint-marginals","seed":202610264033,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, 1/8], [-4, 1/4], [-3, 1/8], [-2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0034","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27e215647c8cd2f830426f5a5ae9908931a55fde7a6627c2dc817fe94bc91b8","family_id":"qf-mod26-joint-marginals","seed":202610264034,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, 1/8], [-3, 1/4], [-2, 1/8], [-1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0035","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc0bf219cd1c20f5a0df842608ecca511e49b8b73065554c93e663eb11281a5","family_id":"qf-mod26-joint-marginals","seed":202610264035,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, 1/8], [-2, 1/4], [-1, 1/8], [0, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0036","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00a5ede282f9c94662726ed476d762820ead0e78d8bb7619dc655a08ec204f6e","family_id":"qf-mod26-joint-marginals","seed":202610264036,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 1/8], [-1, 1/4], [0, 1/8], [1, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0037","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5fb77850e1f5f242957441f924e587916fb091afc2ff737ffa0d8993b9fa13c","family_id":"qf-mod26-joint-marginals","seed":202610264037,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0038","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06dc3261ba4c65179ebac95d40efbee1e5c20f744f20f8f435041d6436d23811","family_id":"qf-mod26-joint-marginals","seed":202610264038,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 1/8], [1, 1/4], [2, 1/8], [3, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0039","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd5a82c46e96c1bc8471896296cc6ac4ab92e340908b534115e020f57e1445f","family_id":"qf-mod26-joint-marginals","seed":202610264039,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 1/8], [2, 1/4], [3, 1/8], [4, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0040","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82c62aee67ebb99fbc3dcc3939f1a61a4d36d9e18641af6b8361d9af20835f22","family_id":"qf-mod26-joint-marginals","seed":202610264040,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, 1/8], [3, 1/4], [4, 1/8], [5, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0041","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e419c3c1f65aa7825317cf19fa14223dcce0ef2a2e132854abca7b51619e832d","family_id":"qf-mod26-joint-marginals","seed":202610264041,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, 1/8], [4, 1/4], [5, 1/8], [6, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0042","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8908a19b0b0d57a904abdbb1f0d62d10cac1642d78bdd5aa06978fd8bb8b8ebd","family_id":"qf-mod26-joint-marginals","seed":202610264042,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, 1/8], [5, 1/4], [6, 1/8], [7, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0043","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"517e4fa6e9366790f3992462370dbe73c4a52d7c21b508192b6cb712f5124d58","family_id":"qf-mod26-joint-marginals","seed":202610264043,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0044","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e2132e3a4ec625a6910fe5e71c6f1d522fcffff033b5a6c207a1e017685fc95","family_id":"qf-mod26-joint-marginals","seed":202610264044,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1/8], [7, 1/4], [8, 1/8], [9, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0045","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80f5f7d1611e449109acb42697275e7f1414fe469e1a014ba226a7b0fe47a022","family_id":"qf-mod26-joint-marginals","seed":202610264045,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 1/8], [8, 1/4], [9, 1/8], [10, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0046","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a017684b3932a5f6f09b96c576771e3f4376a685109d5ce7d6252133c23dde5","family_id":"qf-mod26-joint-marginals","seed":202610264046,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 1/8], [9, 1/4], [10, 1/8], [11, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [2, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0047","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b9965f9a8f7df4f67cd5c6d8a932d6c33e43eda8b225e49b42df842bbed1572","family_id":"qf-mod26-joint-marginals","seed":202610264047,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, 1/8], [10, 1/4], [11, 1/8], [12, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [3, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0048","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54c13afcb105d11d5497fff285b75de6d1f2fbaa917baa08d232a3cf6c45e3eb","family_id":"qf-mod26-joint-marginals","seed":202610264048,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[10, -1], [11, 0], [12, 1], [13, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, 1/8], [11, 1/4], [12, 1/8], [13, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0049","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"985b6aaf8dddb11b9c2f89ff7f108539af0fcb623194f427abca3b10147143d2","family_id":"qf-mod26-joint-marginals","seed":202610264049,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, 1/8], [12, 1/4], [13, 1/8], [14, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [5, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0050","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f387424391aec0a63dcd04a7bb3728e7eec978663befda39f8e5499bdff47cc","family_id":"qf-mod26-joint-marginals","seed":202610264050,"semantic_template":"qf-mod26-joint-marginals/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Aggregate Both Marginals from the Same Qualified Joint Law","stem":"Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.","answer":{"kind":"expression","value":"([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])","explanation":"Sum aligned masses over equal first coordinates and separately over equal second coordinates. Do not multiply the resulting marginals or reconstruct an unprovided joint law. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 1/8], [13, 1/4], [14, 1/8], [15, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [6, 1/2]])."},"skills":["Aggregate Both Marginals from the Same Qualified Joint Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0051","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb7046dd2891c98cd975289e7c7101f5afc922b1291e6d34c2adc94a933972d7","family_id":"qf-mod26-joint-rectangle-event","seed":202610264051,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]],\"x_event\":[-12,-10],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-12, -10], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-12, -10], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -10], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0052","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d74ad1533c0037b29187b2699db4be6b09059d33534528e777fb1af1321d283c","family_id":"qf-mod26-joint-rectangle-event","seed":202610264052,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]],\"x_event\":[-11,-9],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-11, -9], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-11, -9], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -9], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0053","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c6a9d5f79b6926c5267db80c4f351f835cdd3217fbc246b67d42e46ee8853e1","family_id":"qf-mod26-joint-rectangle-event","seed":202610264053,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]],\"x_event\":[-10,-8],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-10, -8], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-10, -8], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -8], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0054","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce2daac1d642fa94e808045bb1e85e2217a9854524bbbcb77415d78daeab5922","family_id":"qf-mod26-joint-rectangle-event","seed":202610264054,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]],\"x_event\":[-9,-7],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-9, -7], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-9, -7], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, -7], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0055","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd40c3dfd7a9fe042749037fa2e5e63a605325842b5da128dbe586f2adea98b1","family_id":"qf-mod26-joint-rectangle-event","seed":202610264055,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]],\"x_event\":[-8,-6],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-8, -6], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-8, -6], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, -6], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0056","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dc000a25dd7d9824663c0ef6e86d7982dd5d9f3a7107c1cedde51397bf2d27b","family_id":"qf-mod26-joint-rectangle-event","seed":202610264056,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]],\"x_event\":[-7,-5],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-7, -5], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-7, -5], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, -5], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0057","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76314edec08c5839a9ca002537e490217de59f2ee2cd638d6256fecc6faff74e","family_id":"qf-mod26-joint-rectangle-event","seed":202610264057,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]],\"x_event\":[-6,-4],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-6, -4], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-6, -4], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -4], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0058","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"075bca997256f15f6b84bb0c692bf88d47dda3bbdba6be4af791cef189282679","family_id":"qf-mod26-joint-rectangle-event","seed":202610264058,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]],\"x_event\":[-5,-3],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-5, -3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-5, -3], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, -3], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0059","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d3f9bd80c99b2bdde81d4721424b97d47dca6e46bb7352010d8d9090e6a28b","family_id":"qf-mod26-joint-rectangle-event","seed":202610264059,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]],\"x_event\":[-4,-2],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-4, -2], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-4, -2], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, -2], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0060","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c07ed6741752cc57823bb6864c739bb218f689bb9756b2edef96d65e66c4451","family_id":"qf-mod26-joint-rectangle-event","seed":202610264060,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]],\"x_event\":[-3,-1],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-3, -1], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-3, -1], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, -1], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0061","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0c68da329f282367fbaa2cd4277b6462139bdbc755eaa13c255dfb471faa509","family_id":"qf-mod26-joint-rectangle-event","seed":202610264061,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"x_event\":[-2,0],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-2, 0], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-2, 0], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 0], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0062","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493b2f7d70f03d6335baf7999dfa317d115f13d971e654577bbd013a11de58a7","family_id":"qf-mod26-joint-rectangle-event","seed":202610264062,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]],\"x_event\":[-1,1],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[-1, 1], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([-1, 1], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, 1], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0063","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada9d138dbc6457788f18df382500a8f04b785948375ab17318e0d59c0360060","family_id":"qf-mod26-joint-rectangle-event","seed":202610264063,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]],\"x_event\":[0,2],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[0, 2], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([0, 2], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 2], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0064","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8a654e331f05077222935b5a6f9f14f41f509aa55ae8c404cf0138e84ad41ee","family_id":"qf-mod26-joint-rectangle-event","seed":202610264064,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]],\"x_event\":[1,3],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[1, 3], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([1, 3], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 3], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0065","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c336a0696f0c6a5ceee43ed370d4222d82f4b5a6c0b1cc813df2abba4d34e8fc","family_id":"qf-mod26-joint-rectangle-event","seed":202610264065,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]],\"x_event\":[2,4],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[2, 4], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([2, 4], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 4], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0066","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7127bc0d665339a4eda5f4876b72f7539080b6d34e5d4281978e66c2838d515a","family_id":"qf-mod26-joint-rectangle-event","seed":202610264066,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]],\"x_event\":[3,5],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[3, 5], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([3, 5], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 5], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0067","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24697771ce8f3b77633753e2d1215d83819299b4e298c303c059f3e4cc6bb467","family_id":"qf-mod26-joint-rectangle-event","seed":202610264067,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]],\"x_event\":[4,6],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[4, 6], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([4, 6], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 6], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0068","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b0574f2bfa363b5dccb0309b6d4e8edebffdc3998f64473b3b21f52ee7daee0","family_id":"qf-mod26-joint-rectangle-event","seed":202610264068,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]],\"x_event\":[5,7],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[5, 7], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([5, 7], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 7], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0069","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38d4d8c4f7196db70382fccb2269c4a401cc2a8ddd1b963b7700a21acff33c79","family_id":"qf-mod26-joint-rectangle-event","seed":202610264069,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]],\"x_event\":[6,8],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[6, 8], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([6, 8], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 8], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0070","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a19b364aa88fa834fd44b482937c48db252759d3934cb409e511ebbbec44073","family_id":"qf-mod26-joint-rectangle-event","seed":202610264070,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]],\"x_event\":[7,9],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[7, 9], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([7, 9], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 9], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0071","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a559a63432de63d37ce741927695d1de6d8ed1e59cf00cf4af27272ce5e66c0","family_id":"qf-mod26-joint-rectangle-event","seed":202610264071,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]],\"x_event\":[8,10],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[8, 10], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([8, 10], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 10], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0072","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec87773d10ac302e3e4df46ef2e0f3bcc75719a109ba518793d351c7c149cc8b","family_id":"qf-mod26-joint-rectangle-event","seed":202610264072,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]],\"x_event\":[9,11],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[9, 11], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([9, 11], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 11], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0073","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"767da1ec53601d0c191fb5c8e75e72608a770102add6881ae5b5d4a3c997c3a8","family_id":"qf-mod26-joint-rectangle-event","seed":202610264073,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]],\"x_event\":[10,12],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[10, -1], [11, 0], [12, 1], [13, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[10, 12], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([10, 12], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 12], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0074","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a825c6faf2f7080737a3f3361fa17027d67d70d7ed59f8f264fbb68de69f1e06","family_id":"qf-mod26-joint-rectangle-event","seed":202610264074,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]],\"x_event\":[11,13],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[11, 13], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([11, 13], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 13], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0075","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7ca3c42c17a10f9303bb680397c3d1fdfe633fa11bdd67fc46a3841b642e9c5","family_id":"qf-mod26-joint-rectangle-event","seed":202610264075,"semantic_template":"qf-mod26-joint-rectangle-event/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]],\"x_event\":[12,14],\"y_event\":[-1,1]}"},"fields":{"title":"Both Coordinates Must Belong to a Rectangle Event","stem":"Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first. x_event=[12, 14], y_event=[-1, 1] are sets: normalize duplicates and numeric order. Give (normalized x_event,normalized y_event,probability). Outside-support labels remain in the normalized sets but add no mass.","answer":{"kind":"expression","value":"([12, 14], [-1, 1], 1/4)","explanation":"Sum the mass of a pair only when its first coordinate belongs to x_event AND its second belongs to y_event. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 14], [-1, 1], 1/4)."},"skills":["Both Coordinates Must Belong to a Rectangle Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0076","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820d541313b45981668999a4f6f3e4683ae183880cac1af1b7bc5c8346f757ef","family_id":"qf-mod26-joint-mass-lookup","seed":202610264076,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-12,-1],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-12, -1], [-11, 0], [-10, 1], [-9, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-12, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-12, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0077","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc5c067f5ecb61d36b2fdd34ee51d2f8f026c5785a40bc851beb6111630efcbb","family_id":"qf-mod26-joint-mass-lookup","seed":202610264077,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-10,0],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-11, -1], [-10, 0], [-9, 1], [-8, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-10, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-10, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0078","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"881f25624f1c47785289893f60c2b6fe2401dc4d18391ed49aac4f3aa0ae147b","family_id":"qf-mod26-joint-mass-lookup","seed":202610264078,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,1],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-10, -1], [-9, 0], [-8, 1], [-7, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-8, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-8, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0079","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"307c56c0fd1a36729f375e619c22e6eb565e97946989ac5189d641dca32410d8","family_id":"qf-mod26-joint-mass-lookup","seed":202610264079,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[3,99],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-9, -1], [-8, 0], [-7, 1], [-6, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[3, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([3, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0080","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf100c97d0a0313cbe0c7900fb99f2d5088d32360c1bcfd9194ed3455933ba91","family_id":"qf-mod26-joint-mass-lookup","seed":202610264080,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-8,-1],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-8, -1], [-7, 0], [-6, 1], [-5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-8, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-8, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0081","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"423a59bc19928e5b4f7058e74ab49c3df68711dbaf28a885d89fc33bd8e71a18","family_id":"qf-mod26-joint-mass-lookup","seed":202610264081,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-6,0],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-7, -1], [-6, 0], [-5, 1], [-4, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-6, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-6, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0082","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7537428daf700b63003add96c431e335cbc4b1fde2280868e1f94ff07615348b","family_id":"qf-mod26-joint-mass-lookup","seed":202610264082,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,1],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-6, -1], [-5, 0], [-4, 1], [-3, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-4, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-4, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0083","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2e2973be0be038d7733444cbb5186baddc904d08c39a34f896475a3562e9ce4","family_id":"qf-mod26-joint-mass-lookup","seed":202610264083,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[7,99],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-5, -1], [-4, 0], [-3, 1], [-2, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[7, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([7, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0084","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2353cd7cdaffe7afdfb0694de8d8020b70686b275f90d4c0085c9d963d213af","family_id":"qf-mod26-joint-mass-lookup","seed":202610264084,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-4,-1],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-4, -1], [-3, 0], [-2, 1], [-1, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-4, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-4, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0085","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95387c85777b27bbfe1a625435ed4f6a8e3bbb94ee22cbca1b46c22c1c050228","family_id":"qf-mod26-joint-mass-lookup","seed":202610264085,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[-2,0],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-3, -1], [-2, 0], [-1, 1], [0, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[-2, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([-2, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0086","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac6af6efae7719eea5d0990831557531d5e630cdb3d954586dd2f84ca72a99d2","family_id":"qf-mod26-joint-mass-lookup","seed":202610264086,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,1],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[0, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([0, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0087","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8edf6920e086c1a8ffceae8902ec43854ea110f59dadbc84b3cfa06a52929a6b","family_id":"qf-mod26-joint-mass-lookup","seed":202610264087,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[11,99],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[-1, -1], [0, 0], [1, 1], [2, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[11, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([11, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0088","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba3f10924accbc584facca34a0f6c4a2625bb129c14485196abdb5361438aca","family_id":"qf-mod26-joint-mass-lookup","seed":202610264088,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[0,-1],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[0, -1], [1, 0], [2, 1], [3, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[0, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([0, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0089","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e0ef99cf259056b456ee57f3b43c23da4e4557ab2f4828766a83070f27b5cff","family_id":"qf-mod26-joint-mass-lookup","seed":202610264089,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[2,0],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[1, -1], [2, 0], [3, 1], [4, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[2, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([2, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0090","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"035ac5b40d6eec3dbe321b114f40b08a9d6c96361375fba81a0c11c34b10a425","family_id":"qf-mod26-joint-mass-lookup","seed":202610264090,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,1],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[2, -1], [3, 0], [4, 1], [5, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[4, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([4, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0091","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ec94beb9de3e0a8a5723910c77fd5bcae92831a4eafe6578f3c1b2adab74969","family_id":"qf-mod26-joint-mass-lookup","seed":202610264091,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[15,99],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[3, -1], [4, 0], [5, 1], [6, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[15, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([15, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([15, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0092","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aab7558e1f211df837d47d6832c6e875e521854feb5cd3a97758803cb21aa3d","family_id":"qf-mod26-joint-mass-lookup","seed":202610264092,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[4,-1],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[4, -1], [5, 0], [6, 1], [7, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[4, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([4, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0093","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad9152ab27d37438a36869dd2527bc50d660c7dbbb183c0c0634d6e46d531b50","family_id":"qf-mod26-joint-mass-lookup","seed":202610264093,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[6,0],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[6, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([6, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0094","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8595140ce95d6f44c64d207f3e6d58cd71cea95575e76a47a6b0437ee3b34849","family_id":"qf-mod26-joint-mass-lookup","seed":202610264094,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,1],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[8, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([8, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0095","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7647d14e4ceaf707b93aff58a50879da4e60d931d306fd03ea82a0e752f16754","family_id":"qf-mod26-joint-mass-lookup","seed":202610264095,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[19,99],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[7, -1], [8, 0], [9, 1], [10, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[19, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([19, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([19, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0096","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b0a1a409633d29fedf2e3996838b82e975551a2eec3ad47a1f4bab93f812e1","family_id":"qf-mod26-joint-mass-lookup","seed":202610264096,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[8,-1],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[8, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([8, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0097","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73f547d3c8cb93ca9c4e88a9263be831535308ff40f127b64a94d85ff4b5e283","family_id":"qf-mod26-joint-mass-lookup","seed":202610264097,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[10,0],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[9, -1], [10, 0], [11, 1], [12, 3]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[10, 0]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([10, 0], 1/4)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 0], 1/4)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0098","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c96bdd5efd7434f67c2d782427a77ed7c1c97d4a51d053d86d7e75625cd17d7f","family_id":"qf-mod26-joint-mass-lookup","seed":202610264098,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,1],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[10, -1], [11, 0], [12, 1], [13, 4]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[12, 1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([12, 1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0099","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e63442e00717517f3df5053daba6978b78cd239c71523d0540e7145011b29c46","family_id":"qf-mod26-joint-mass-lookup","seed":202610264099,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[23,99],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[11, -1], [12, 0], [13, 1], [14, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[23, 99]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([23, 99], 0)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([23, 99], 0)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r04-finite-joint-distribution-0100","batch_id":"batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9389bd3c6cd0b1fb28f60e1de9e9727f7a297f85062d83884df4580a3df405d","family_id":"qf-mod26-joint-mass-lookup","seed":202610264100,"semantic_template":"qf-mod26-joint-mass-lookup/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"pair\":[12,-1],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Look Up One Pair without Assuming a Product Law","stem":"Use the complete list of distinct integer pairs=[[12, -1], [13, 0], [14, 1], [15, 6]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. The joint table must qualify first; requested pair=[12, -1]. Give (pair,mass), using zero when the pair is unlisted.","answer":{"kind":"expression","value":"([12, -1], 1/8)","explanation":"Use its mass in this exact joint table, not a product of marginal masses. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, -1], 1/8)."},"skills":["Look Up One Pair without Assuming a Product Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}