{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r05-covariance-0001","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"860c288b94cf1a7a6aa937b08ef656c0cfc08c06d6e035494e1a0362f4530b6c","family_id":"qf-mod26-covariance-compute","seed":202610265001,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-10, 1, -35/4, 5/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, -35/4, 5/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0002","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9c1005470f8faf3a9b2b0d155c2fbcee5d79d1221e7ecc8f087b02eebdb57a4","family_id":"qf-mod26-covariance-compute","seed":202610265002,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-9, 3/2, -47/4, 7/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-9, 3/2, -47/4, 7/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0003","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac754857811c8bc6a699faa55f29e66f94e58efa516f294427c7c67db3b141bf","family_id":"qf-mod26-covariance-compute","seed":202610265003,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-8, 2, -55/4, 9/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-8, 2, -55/4, 9/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0004","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8139fa81b4992ad03465050780a236c216dd1b56d0c4ef88fe9548b81c90df9","family_id":"qf-mod26-covariance-compute","seed":202610265004,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-7, 5/2, -59/4, 11/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-7, 5/2, -59/4, 11/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0005","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50f1930bafa4dd6f6b5d9bbf719fd3e58755de7e344f7a9ae8d4990490cf357f","family_id":"qf-mod26-covariance-compute","seed":202610265005,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-6, 3, -59/4, 13/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-6, 3, -59/4, 13/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0006","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c80832fa0a854787d83cfa1725557dffb2fa2bde0a38fd0f1b6b6f37c44c5aa2","family_id":"qf-mod26-covariance-compute","seed":202610265006,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-5, 1, -15/4, 5/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, -15/4, 5/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0007","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"201d9a3f2d94a91d2f81a1cec4dc5474795b0573c99356ee85c161cb7db3cdf1","family_id":"qf-mod26-covariance-compute","seed":202610265007,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-4, 3/2, -17/4, 7/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-4, 3/2, -17/4, 7/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0008","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"299beb0a9fcb301bb434e656f9bbae9ee11fd4468547d82c9ca4afef2c763416","family_id":"qf-mod26-covariance-compute","seed":202610265008,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-3, 2, -15/4, 9/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 2, -15/4, 9/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0009","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f82b67d8149dcd2ea18448b2c745af701d6f16d4d035b23599b18c658c3adf","family_id":"qf-mod26-covariance-compute","seed":202610265009,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-2, 5/2, -9/4, 11/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 5/2, -9/4, 11/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0010","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2078f9aff2b5cfc3de890dd279d944032a28df854b1dd4dacd1fcb40fb5c710","family_id":"qf-mod26-covariance-compute","seed":202610265010,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(-1, 3, 1/4, 13/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, 1/4, 13/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0011","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5809d0666210cb444e4b8ef1b1461e31b8d8f8635fadf18fb6ae4a2cf72a811c","family_id":"qf-mod26-covariance-compute","seed":202610265011,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(0, 1, 5/4, 5/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, 5/4, 5/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0012","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad0dc187884ef04aa52ca54db06c48238995f96662d6e2297791c0a48dfe8004","family_id":"qf-mod26-covariance-compute","seed":202610265012,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(1, 3/2, 13/4, 7/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3/2, 13/4, 7/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0013","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8779f6c047b06761641cd5e11dae406c786e6ad3745f864d03d484c375016587","family_id":"qf-mod26-covariance-compute","seed":202610265013,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(2, 2, 25/4, 9/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, 25/4, 9/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0014","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b41003a09578da7993eb0e35702223544ea4e74dc9893fcd95fe927c87363b39","family_id":"qf-mod26-covariance-compute","seed":202610265014,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(3, 5/2, 41/4, 11/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 5/2, 41/4, 11/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0015","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf80c81e2e66279eac046fe7b6546de229b1b7e839d63fc55d707b3696a7ea3","family_id":"qf-mod26-covariance-compute","seed":202610265015,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(4, 3, 61/4, 13/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 3, 61/4, 13/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0016","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43d9645fd7d3e5b856369eb4f580084ab3912f916b824dd50d42fa2c8da5449","family_id":"qf-mod26-covariance-compute","seed":202610265016,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(5, 1, 25/4, 5/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, 25/4, 5/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0017","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79dd9471381cd39bc9214c7ad293c98e1c567fbe90e4ea1a1e0ae64db39d8d36","family_id":"qf-mod26-covariance-compute","seed":202610265017,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(6, 3/2, 43/4, 7/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 3/2, 43/4, 7/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0018","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84493223b13ae0cf8596fc9c412513905ef587a1c2a5bd8464f6a0df21e64ee","family_id":"qf-mod26-covariance-compute","seed":202610265018,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(7, 2, 65/4, 9/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 2, 65/4, 9/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0019","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dcbf97b2c9219c4e067b35934c7dac6020c02931df994714b1b7791b63e8f34","family_id":"qf-mod26-covariance-compute","seed":202610265019,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(8, 5/2, 91/4, 11/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 5/2, 91/4, 11/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0020","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f0bcdf0f2cf31902ca21a314d070567445de42f4eb9e150808d855e11e19feb","family_id":"qf-mod26-covariance-compute","seed":202610265020,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(9, 3, 121/4, 13/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 3, 121/4, 13/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0021","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f7356a946490a0c253141e036298f3d82a68a938f6929618e70733adbb9905c","family_id":"qf-mod26-covariance-compute","seed":202610265021,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(10, 1, 45/4, 5/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. 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, 45/4, 5/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0022","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8f53e47351fe8f347aa30d01b367214e0831d62005cb220bc3f4f04310478c1","family_id":"qf-mod26-covariance-compute","seed":202610265022,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(11, 3/2, 73/4, 7/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 3/2, 73/4, 7/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0023","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068918d42a958f59dfac1c668d9a7c6a5b21566756ab72cd1dc04d7be0da49b1","family_id":"qf-mod26-covariance-compute","seed":202610265023,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(12, 2, 105/4, 9/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 2, 105/4, 9/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0024","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4075d1d038d868df94fc9a7bb844b0cf8042610b1b4f63415adc95194606d7fd","family_id":"qf-mod26-covariance-compute","seed":202610265024,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(13, 5/2, 141/4, 11/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 5/2, 141/4, 11/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0025","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"698a0893807db67384ed1d64e8357ff9c3712e3acc0be22d2a9b60c5061b9788","family_id":"qf-mod26-covariance-compute","seed":202610265025,"semantic_template":"qf-mod26-covariance-compute/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Compute Three Moments from One 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. Require a qualified joint table. Give (E_X,E_Y,E_XY,covariance). Cov(X,Y)=E[XY]−E[X]E[Y].","answer":{"kind":"expression","value":"(14, 3, 181/4, 13/4)","explanation":"Compute all three expectations with the same aligned masses before subtracting. Mixing expectations from different models is invalid. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 3, 181/4, 13/4)."},"skills":["Compute Three Moments from One 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-r05-covariance-0026","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bff42463a229933e7b3c936ecd6f40568fdedbfdd1bddafd2f15b7683d18f279","family_id":"qf-mod26-covariance-symmetry","seed":202610265026,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(5/4, 5/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0027","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7efb3d33a6b79703842b143c8d191cbb00c2dae6352dc7dd3606af8e042eb79a","family_id":"qf-mod26-covariance-symmetry","seed":202610265027,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(7/4, 7/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0028","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2131db0fde27100ef72a8cdafc7d78af4e717eae6d1e7d73580aaf80e3c04e5a","family_id":"qf-mod26-covariance-symmetry","seed":202610265028,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(9/4, 9/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0029","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9053ec177c9e321dd33e6e616cb9ceff92618ff46cca118f6d64ea64fdb76686","family_id":"qf-mod26-covariance-symmetry","seed":202610265029,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(11/4, 11/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0030","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d42ecd4e35e740e674924d83d4eaa05e88182a9fa18abecc59486d7e3e6603ca","family_id":"qf-mod26-covariance-symmetry","seed":202610265030,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(13/4, 13/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0031","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f07aa8c8f798d9781c107617a0a54c0e99a4f65082157c8a4f6e04e6d6c9459","family_id":"qf-mod26-covariance-symmetry","seed":202610265031,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(5/4, 5/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0032","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84a72634d9725b2d1f47a591aaf9f0a44f6b08da22b326648cceaf7b9c733378","family_id":"qf-mod26-covariance-symmetry","seed":202610265032,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(7/4, 7/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0033","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"616ff7fb25fe2443868d87352b49e77100f7667a5cfd98fa25eed8e2aabef04d","family_id":"qf-mod26-covariance-symmetry","seed":202610265033,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(9/4, 9/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0034","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6017e07742057e3bd7bd909c381697897f0c36ed65d96382c3b914929d889f","family_id":"qf-mod26-covariance-symmetry","seed":202610265034,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(11/4, 11/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0035","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0df05b6f321cafd7f398cccda9e9ee489911de85adb42346c51fca70462ca8c6","family_id":"qf-mod26-covariance-symmetry","seed":202610265035,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(13/4, 13/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0036","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15ffc3b428f006efc6917be6660becf0a03ce04c79ea4d570a671e35ea1de510","family_id":"qf-mod26-covariance-symmetry","seed":202610265036,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(5/4, 5/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0037","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1405bf43d6a394dc4fce3922da2aa56872d64e053fee5d1613cded320d6a35","family_id":"qf-mod26-covariance-symmetry","seed":202610265037,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(7/4, 7/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0038","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b146e6362195983ff5ecd445875772e02efd4155ee550593faf56caafd6ffd94","family_id":"qf-mod26-covariance-symmetry","seed":202610265038,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(9/4, 9/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0039","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3049335d0073953cf018632f377f9313106f431f42567c7585f66ee6e270f3ff","family_id":"qf-mod26-covariance-symmetry","seed":202610265039,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(11/4, 11/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0040","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6adaf870cbe93a43ff9af79902f0ddd35b08f61dd073d29b2a36b44ee4e82a4","family_id":"qf-mod26-covariance-symmetry","seed":202610265040,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(13/4, 13/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0041","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f770db83dac19b37c9b04fd21d9b1cb34fa37afa4cb4f0a427bc323f402af5a9","family_id":"qf-mod26-covariance-symmetry","seed":202610265041,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(5/4, 5/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0042","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44b026d9424617b39a0e03ec9825c6ef50bf6525e0711415440c2adbd72bca08","family_id":"qf-mod26-covariance-symmetry","seed":202610265042,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(7/4, 7/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0043","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f0806ba296afe7e7ffeb07dc3adff15423ca6abf5a7e34f86096d0d7d5f4527","family_id":"qf-mod26-covariance-symmetry","seed":202610265043,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(9/4, 9/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0044","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b62cf125380ad76dd154a6646481604fc72f5f0ebaca56023369fc730564ebcb","family_id":"qf-mod26-covariance-symmetry","seed":202610265044,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(11/4, 11/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0045","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"758a9d4e0b6a7ba8e964f0e0440dbb63f5434cd891f2fe562ff06f012b88aaaf","family_id":"qf-mod26-covariance-symmetry","seed":202610265045,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(13/4, 13/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0046","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f4f097238afb4ad9ab89d80a2c3435556a7e33b1bd19f4362e0a75873c9e2f2","family_id":"qf-mod26-covariance-symmetry","seed":202610265046,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(5/4, 5/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/4, 5/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0047","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9b16dc81cb135fe388ee388198fc7dcac3f345855d5fab3f2574601a080a93a","family_id":"qf-mod26-covariance-symmetry","seed":202610265047,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(7/4, 7/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7/4, 7/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0048","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e52703e3e81cb1852b7003a0a43cd2cdd0aed9bc36c0b297395c3c63798ede9e","family_id":"qf-mod26-covariance-symmetry","seed":202610265048,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(9/4, 9/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9/4, 9/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0049","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"848d34820ed3a87d4904cf7516b919fd6f246c2c89593770d869e991d85d417d","family_id":"qf-mod26-covariance-symmetry","seed":202610265049,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(11/4, 11/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11/4, 11/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0050","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19833a4f017e9f60266a8fe529272b3e867336b98680a68f929ace3880347483","family_id":"qf-mod26-covariance-symmetry","seed":202610265050,"semantic_template":"qf-mod26-covariance-symmetry/{\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Symmetry of Covariance under Swapping Coordinates","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. Require a qualified joint table. Give (Cov_XY,Cov_YX,equal?).","answer":{"kind":"expression","value":"(13/4, 13/4, Yes)","explanation":"Multiplication of the coordinates commutes and the product of their expectations commutes, so swapping both roles preserves covariance. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13/4, 13/4, Yes)."},"skills":["Symmetry of Covariance under Swapping Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0051","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09e5fa3050c7e50844596f261ad783dbd81fea158b9d28b40514bee36af3b0be","family_id":"qf-mod26-covariance-affine","seed":202610265051,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-3,\"b\":-12,\"c\":-2,\"d\":12,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-12,-1],[-11,0],[-10,1],[-9,2]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-3, -12, -2, 12]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-3, -12, -2, 12, 5/4, 15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, -12, -2, 12, 5/4, 15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0052","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0041d92748b98aa8e302af3cd8247539392aad3d99fca7ab726a206f450a564a","family_id":"qf-mod26-covariance-affine","seed":202610265052,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-2,\"b\":-11,\"c\":-1,\"d\":11,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-11,-1],[-10,0],[-9,1],[-8,3]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-2, -11, -1, 11]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-2, -11, -1, 11, 7/4, 7/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -11, -1, 11, 7/4, 7/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0053","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc28f50566bb7d46616a7cd6a07ce51419d715396e0293bdfa49fe5b639f0396","family_id":"qf-mod26-covariance-affine","seed":202610265053,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-1,\"b\":-10,\"c\":0,\"d\":10,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-10,-1],[-9,0],[-8,1],[-7,4]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-1, -10, 0, 10]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-1, -10, 0, 10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -10, 0, 10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0054","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b19e4b48ce4f15388ecc9cc317aceb07d2f38eab7945cef92ccc930670bb5e55","family_id":"qf-mod26-covariance-affine","seed":202610265054,"semantic_template":"qf-mod26-covariance-affine/{\"a\":0,\"b\":-9,\"c\":1,\"d\":9,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-9,-1],[-8,0],[-7,1],[-6,5]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, -9, 1, 9]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(0, -9, 1, 9, 11/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -9, 1, 9, 11/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0055","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8dde9aa162f481f328d7612e4a1a0c1b0d9537090262bd69cac40d7f9270f3","family_id":"qf-mod26-covariance-affine","seed":202610265055,"semantic_template":"qf-mod26-covariance-affine/{\"a\":1,\"b\":-8,\"c\":2,\"d\":8,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-8,-1],[-7,0],[-6,1],[-5,6]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[1, -8, 2, 8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(1, -8, 2, 8, 13/4, 13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -8, 2, 8, 13/4, 13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0056","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"208351d34bba483cd9d6ea7b33e04395b0b1c1a76443a24561cda03542bb9f25","family_id":"qf-mod26-covariance-affine","seed":202610265056,"semantic_template":"qf-mod26-covariance-affine/{\"a\":2,\"b\":-7,\"c\":-2,\"d\":7,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-7,-1],[-6,0],[-5,1],[-4,2]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[2, -7, -2, 7]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(2, -7, -2, 7, 5/4, -5, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -7, -2, 7, 5/4, -5, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0057","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58cc9e68579d1a15e0427cd90d1f109ce554d1f4ca70210d962682225f3f9ad8","family_id":"qf-mod26-covariance-affine","seed":202610265057,"semantic_template":"qf-mod26-covariance-affine/{\"a\":3,\"b\":-6,\"c\":-1,\"d\":6,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-6,-1],[-5,0],[-4,1],[-3,3]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[3, -6, -1, 6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(3, -6, -1, 6, 7/4, -21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, -6, -1, 6, 7/4, -21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0058","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e4d48d0b6b840b943c28fcada5a9a0403dbd83151fe9ea08969c5a5d7aa034e","family_id":"qf-mod26-covariance-affine","seed":202610265058,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-3,\"b\":-5,\"c\":0,\"d\":5,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-5,-1],[-4,0],[-3,1],[-2,4]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-3, -5, 0, 5]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-3, -5, 0, 5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 0, 5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0059","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa36db094a71fab2242ba8534c071b6c501b34e970c30762a1283912fcb1938","family_id":"qf-mod26-covariance-affine","seed":202610265059,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-2,\"b\":-4,\"c\":1,\"d\":4,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-4,-1],[-3,0],[-2,1],[-1,5]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-2, -4, 1, 4]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-2, -4, 1, 4, 11/4, -11/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 4, 11/4, -11/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0060","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257e2af98374d902b1da4840ed7a45620bda316cef6282956bda2b6a8d62631a","family_id":"qf-mod26-covariance-affine","seed":202610265060,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-1,\"b\":-3,\"c\":2,\"d\":3,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-3,-1],[-2,0],[-1,1],[0,6]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-1, -3, 2, 3]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-1, -3, 2, 3, 13/4, -13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 2, 3, 13/4, -13/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0061","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19bfae2c0461d8346dc0bb96cb8ba82e3be4e57307030ee7125c1d69e420baeb","family_id":"qf-mod26-covariance-affine","seed":202610265061,"semantic_template":"qf-mod26-covariance-affine/{\"a\":0,\"b\":-2,\"c\":-2,\"d\":2,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, -2, -2, 2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0062","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2a551be521dac42d0923ec584866694fa9df88f67592e47fec5ce48fcac2c6","family_id":"qf-mod26-covariance-affine","seed":202610265062,"semantic_template":"qf-mod26-covariance-affine/{\"a\":1,\"b\":-1,\"c\":-1,\"d\":1,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-1,-1],[0,0],[1,1],[2,3]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[1, -1, -1, 1]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(1, -1, -1, 1, 7/4, -7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 7/4, -7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0063","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5340e9e6da512dc3b17b12ac3952a365eec15457323621b64039e7dce7cd71a4","family_id":"qf-mod26-covariance-affine","seed":202610265063,"semantic_template":"qf-mod26-covariance-affine/{\"a\":2,\"b\":0,\"c\":0,\"d\":0,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[0,-1],[1,0],[2,1],[3,4]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[2, 0, 0, 0]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(2, 0, 0, 0, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 0, 0, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0064","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b66fe988eb1be111ac3aabaed8c5d991d2f6edb981e523f17ffe39a43f6d2b","family_id":"qf-mod26-covariance-affine","seed":202610265064,"semantic_template":"qf-mod26-covariance-affine/{\"a\":3,\"b\":1,\"c\":1,\"d\":-1,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[1,-1],[2,0],[3,1],[4,5]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[3, 1, 1, -1]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(3, 1, 1, -1, 11/4, 33/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. 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, 11/4, 33/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0065","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0c90c934ab723b0c8143bc3bf0de48221a5bcb0b65480a8f4b30e71c60e436","family_id":"qf-mod26-covariance-affine","seed":202610265065,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-3,\"b\":2,\"c\":2,\"d\":-2,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[2,-1],[3,0],[4,1],[5,6]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-3, 2, 2, -2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-3, 2, 2, -2, 13/4, -39/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 2, 2, -2, 13/4, -39/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0066","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2687021539a59a707675ada1fec3b175d5cb4be2c56f33d3aa0b5c37656da517","family_id":"qf-mod26-covariance-affine","seed":202610265066,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-2,\"b\":3,\"c\":-2,\"d\":-3,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[3,-1],[4,0],[5,1],[6,2]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-2, 3, -2, -3]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-2, 3, -2, -3, 5/4, 5, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, -2, -3, 5/4, 5, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0067","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3015e9c8d357bcaa0de7f6055fc7bbb467b7534dafe82e8aec0c16cb8794e9","family_id":"qf-mod26-covariance-affine","seed":202610265067,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-1,\"b\":4,\"c\":-1,\"d\":-4,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[4,-1],[5,0],[6,1],[7,3]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-1, 4, -1, -4]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-1, 4, -1, -4, 7/4, 7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 4, -1, -4, 7/4, 7/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0068","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40d49fc4f63e9fd71d7894b4a717feb33830ef8faff102d8c3fee4c8a7f1a269","family_id":"qf-mod26-covariance-affine","seed":202610265068,"semantic_template":"qf-mod26-covariance-affine/{\"a\":0,\"b\":5,\"c\":0,\"d\":-5,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[5,-1],[6,0],[7,1],[8,4]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, 5, 0, -5]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(0, 5, 0, -5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 5, 0, -5, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0069","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1976e619a26fcd89882baa8ce76d2280aeb5dd3dd9d6f895435d135aba4a79cc","family_id":"qf-mod26-covariance-affine","seed":202610265069,"semantic_template":"qf-mod26-covariance-affine/{\"a\":1,\"b\":6,\"c\":1,\"d\":-6,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[6,-1],[7,0],[8,1],[9,5]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[1, 6, 1, -6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(1, 6, 1, -6, 11/4, 11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 6, 1, -6, 11/4, 11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0070","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd96534c3fc61e518e1caba4733ecf7a4738044422a8fe57a0c10b14b3da95c4","family_id":"qf-mod26-covariance-affine","seed":202610265070,"semantic_template":"qf-mod26-covariance-affine/{\"a\":2,\"b\":7,\"c\":2,\"d\":-7,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[7,-1],[8,0],[9,1],[10,6]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[2, 7, 2, -7]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(2, 7, 2, -7, 13/4, 13, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 7, 2, -7, 13/4, 13, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0071","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfab7fe69011e374f633a7362db3d62c10ff17c91089623fa64df089c45b603a","family_id":"qf-mod26-covariance-affine","seed":202610265071,"semantic_template":"qf-mod26-covariance-affine/{\"a\":3,\"b\":8,\"c\":-2,\"d\":-8,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[8,-1],[9,0],[10,1],[11,2]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[3, 8, -2, -8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(3, 8, -2, -8, 5/4, -15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 8, -2, -8, 5/4, -15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0072","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38dae29b3a1c3b47decf72455f6da2a1ea8e3ba614da3f3f9fa2101fba089d4e","family_id":"qf-mod26-covariance-affine","seed":202610265072,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-3,\"b\":9,\"c\":-1,\"d\":-9,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[9,-1],[10,0],[11,1],[12,3]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-3, 9, -1, -9]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-3, 9, -1, -9, 7/4, 21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-3, 9, -1, -9, 7/4, 21/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0073","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa0635354b6c9e46f27ffae55592205c1221fa81d283bfda08ef1c8893ad6a6","family_id":"qf-mod26-covariance-affine","seed":202610265073,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-2,\"b\":10,\"c\":0,\"d\":-10,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[10,-1],[11,0],[12,1],[13,4]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-2, 10, 0, -10]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-2, 10, 0, -10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 10, 0, -10, 9/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0074","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da756f607b5c51cd49158d86319df52796c24149e17a2619cf96d09bec5fce5","family_id":"qf-mod26-covariance-affine","seed":202610265074,"semantic_template":"qf-mod26-covariance-affine/{\"a\":-1,\"b\":11,\"c\":1,\"d\":-11,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[11,-1],[12,0],[13,1],[14,5]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[-1, 11, 1, -11]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(-1, 11, 1, -11, 11/4, -11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 11, 1, -11, 11/4, -11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0075","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afe9c895e9d92138138946ac459baed540d779b55b308b2aae4efef6a629119a","family_id":"qf-mod26-covariance-affine","seed":202610265075,"semantic_template":"qf-mod26-covariance-affine/{\"a\":0,\"b\":12,\"c\":2,\"d\":-12,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[12,-1],[13,0],[14,1],[15,6]]}"},"fields":{"title":"Affine Shifts Drop Out of Covariance","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. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, 12, 2, -12]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).","answer":{"kind":"expression","value":"(0, 12, 2, -12, 13/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))","explanation":"The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 12, 2, -12, 13/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))."},"skills":["Affine Shifts Drop Out of Covariance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0076","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c71ddaa6d92785dc6fb5d7399ed9946466cff1a237223189f92ac60efa826f07","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265076,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":1}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=1, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(1, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0077","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c62026386685f7022b2178cde4d94bddc9bfe6bad90906da3372ce3c0aa2d7f","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265077,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":2}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=2, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(2, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. 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, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0078","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ca952e81381f47af855731366928e20d4a3102a3f8f03f3e7f86bf99984006","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265078,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":3}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=3, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(3, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0079","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada7bf6324c5993d609c3b069c33edbebcc2a84afd6c515eafd6f61cde71be9f","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265079,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":4}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=4, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(4, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0080","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45223e7bfd89ad3c6fdc03d0f805cc7ea3dc3fb59dc7da94f7205fb17307df17","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265080,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":5}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=5, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(5, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0081","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47cd7a3a2c0900819d8a54bc6e94c1236081a75224ebe8d1eb5cc29b9f67ae77","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265081,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":6}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=6, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(6, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. 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, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0082","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eefb611f0d3e55af2d288c5de94c0890c0d4759dcc6ae542ad55111ff4664925","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265082,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":7}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=7, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(7, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0083","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c548d51595a3516c808ed95b91aaa6ba75bea54ff6c175a2cc0ed87d02f73182","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265083,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":8}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=8, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(8, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0084","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ef3790e3a5893336cead7fae66d01fcfc74763e2a7814738eb77fea7a4f863b","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265084,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":9}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=9, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(9, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0085","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57a6447df566771900f6769e9b7d777eaad935f68117a84e87fc2ec302e7589b","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265085,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":10}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=10, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(10, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. 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, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0086","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2210d9da03a2e128590af9538edc7ac5afd8bd1517f8ac1b4c89f8a9c60392d","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265086,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":11}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=11, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(11, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0087","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8a37f59c9e9bfda0f4d12505ec42c08e29314fef9bf46e538bc52127a9319c0","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265087,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":12}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=12, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(12, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0088","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5905137bb51880815282fc20925ddb88d7836fb8de117affc9d23fc1f279019","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265088,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":13}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=13, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(13, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0089","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564e802c7d7f4c447db49a2c102ce01d1904e5c03aa09587e9217b2e781a0cd2","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265089,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":14}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=14, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(14, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0090","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62dbfe345e7f56f9a04ad780e184ed30b9cbc42791572b5247b24da130047ecd","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265090,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":15}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=15, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(15, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (15, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0091","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94fff890b0dc1f3eff77bf014871965378c682ce081eb334bf3d94ec14abdf57","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265091,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":16}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=16, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(16, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (16, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0092","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"603fb90f715743e4c790e15f477ee7d8831ac1f4612064c34f1915fe285d1fab","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265092,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":17}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=17, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(17, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (17, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0093","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9ef5a217e7d1d2436ece7b96fa30dcf26e840e0c5bfd35cd1a69d9d1846d274","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265093,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":18}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=18, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(18, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (18, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0094","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07db9d00b4ad93ff7020b93d4cb364f5bcb8a6300cfcde04679b8166888126c9","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265094,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":19}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=19, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(19, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (19, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0095","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97195207c3396d453df5b4530b4f6bc81253878f1e6da420e853949a5d9f369c","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265095,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":20}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=20, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(20, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (20, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0096","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4518c3b988e2639a51d68205fc218f7cdcadb44299057ec14acd198846ffc5f","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265096,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":21}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=21, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(21, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (21, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0097","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afdd85fffd6d1b1d1a98b216f9ccdd84fa3f38f256621c11681540426d9fe04f","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265097,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":22}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=22, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(22, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (22, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0098","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8b50c680a0267c35a32efb87db16bca6824a1d46333d5bd3876448e2edf4ad2","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265098,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":23}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=23, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(23, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (23, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0099","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d856ad81eedf85d491d59538beced3dc0b888d89fb7db0fa7a81b52ed16121d3","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265099,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":24}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=24, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(24, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (24, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r05-covariance-0100","batch_id":"batch-20261013-advance-20261026-r05-covariance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eb1c0eba201184f6e31d779ffde1f44d5b7f4977ee056ec05d7c61d884b0ec5","family_id":"qf-mod26-covariance-zero-counterexample","seed":202610265100,"semantic_template":"qf-mod26-covariance-zero-counterexample/{\"scale\":25}"},"fields":{"title":"Zero Covariance Can Coexist with Dependence","stem":"Take X uniformly on [−scale,0,scale] for nonzero positive scale=25, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.","answer":{"kind":"expression","value":"(25, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)","explanation":"The symmetric odd moment and E[X] vanish, giving zero covariance. But the joint zero event has mass 1/3 while the product of its marginal masses is 1/9. Thus this example is dependent. Nonzero covariance can refute independence; zero covariance alone cannot establish it. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (25, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)."},"skills":["Zero Covariance Can Coexist with Dependence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}