{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0001","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbc16bb50d73fdd472e2302209c1d520eb7366580fa8f7f1a8dfca24e7749123","family_id":"qf-mod28-rvind-qualification","seed":202610286001,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-12,13],\"y_values\":[-3,20]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], No)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], No)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0002","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00dd6bb2edc40b8d15685e16df73d29456db6430d53a2ebee12dfa3df674f04f","family_id":"qf-mod28-rvind-qualification","seed":202610286002,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_values\":[-11,14],\"y_values\":[-2,21]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0003","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07471d5f37ba9bcd3331a8277b7ba297527d28cc3e95ebe93450eccad86ee77d","family_id":"qf-mod28-rvind-qualification","seed":202610286003,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_values\":[-10,15],\"y_values\":[-1,22]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0004","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4de2be2e12cf6a82c5073125d3887357a32c71d3d50ec57c395a23c8b0b59286","family_id":"qf-mod28-rvind-qualification","seed":202610286004,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_values\":[-9,16],\"y_values\":[0,23]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0005","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61af2494c94debe53fbc5dd52f2fb33f7c40b4a905e6df603f9200c752ba3fa5","family_id":"qf-mod28-rvind-qualification","seed":202610286005,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_values\":[-8,17],\"y_values\":[1,24]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0006","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44818bae974ae79cf90cbc76973842bc63d13ef16ebe98d63c378cc34d161f85","family_id":"qf-mod28-rvind-qualification","seed":202610286006,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-7,18],\"y_values\":[2,25]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], No)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], No)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0007","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"392b011a727406b1b1e3aec9185c28b16ada585ce18f7bc7f34a6c1235486410","family_id":"qf-mod28-rvind-qualification","seed":202610286007,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_values\":[-6,19],\"y_values\":[3,26]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0008","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6133f21d387a07ea0f85823a74e1b74c41e9a0ac92a4f0ec107eddc74e921276","family_id":"qf-mod28-rvind-qualification","seed":202610286008,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0009","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f518c1865b3b60c91f0a04ca94d2597ff62ecd60308f92bd692716d6f88ce3c1","family_id":"qf-mod28-rvind-qualification","seed":202610286009,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_values\":[-4,21],\"y_values\":[-2,28]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0010","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59b290a76fa1509876c34b42ce809d5c0ce1b348f240f3efbaf8b4bf6923afe1","family_id":"qf-mod28-rvind-qualification","seed":202610286010,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_values\":[-3,22],\"y_values\":[-1,29]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0011","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e31c31a1b55f65b82458ba0903befa007047eccd69ee45b449e702b74dbc5e","family_id":"qf-mod28-rvind-qualification","seed":202610286011,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-2,23],\"y_values\":[0,30]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], No)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], No)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0012","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7cb15a9b3bfcf3d241439df2c5c3e9d0794e7f9684fdea083bdcf328fb17a5e","family_id":"qf-mod28-rvind-qualification","seed":202610286012,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_values\":[-1,24],\"y_values\":[1,31]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0013","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c5dcd5bb26980f67ce4f49bf4e5aef1bb2473a0cf8b36a776dc8ede0a96f84","family_id":"qf-mod28-rvind-qualification","seed":202610286013,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_values\":[0,25],\"y_values\":[2,32]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0014","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"018e028ca3354a0c72c9829b7c9a1caf4f5c826d58639a51e7c8da81fc810b20","family_id":"qf-mod28-rvind-qualification","seed":202610286014,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_values\":[1,26],\"y_values\":[3,33]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0015","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f8ac54c0d8ffa6b55f2b7c706a247e482880667d9e01613a028d2947ea8ed5c","family_id":"qf-mod28-rvind-qualification","seed":202610286015,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_values\":[2,27],\"y_values\":[-3,34]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0016","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ffde97ea8986fd5df45702131bc589cb3abde9e11cc0f26cc14482be221488","family_id":"qf-mod28-rvind-qualification","seed":202610286016,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[3,28],\"y_values\":[-2,35]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], No)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], No)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0017","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d2e8af83526ca123a247f0b9aff87992a608bf910302ba11f3d87bde4b0e2b4","family_id":"qf-mod28-rvind-qualification","seed":202610286017,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_values\":[4,29],\"y_values\":[-1,36]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0018","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ecaac209c0c46a307f70228c04fcd6602db708b52b52d607ca37b8e772f1562","family_id":"qf-mod28-rvind-qualification","seed":202610286018,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_values\":[5,30],\"y_values\":[0,37]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0019","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e6c6a4b1d8a9141aa36b15a8392f613fbc8394273ee195f0e9afd0d9b8b3efe","family_id":"qf-mod28-rvind-qualification","seed":202610286019,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_values\":[6,31],\"y_values\":[1,38]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0020","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b7a272c412186193dc622601fd8dfac11c1d473a49a0df69091208cad2b8ff2","family_id":"qf-mod28-rvind-qualification","seed":202610286020,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_values\":[7,32],\"y_values\":[2,39]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0021","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1feef2ab7c3c42db0e8cf6c39c818a18c9bcbca9f17d0ab3bc977e30977f0ed","family_id":"qf-mod28-rvind-qualification","seed":202610286021,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[8,33],\"y_values\":[3,40]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[1/2, 0/1], [0/1, 1/2]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], No)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], No)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0022","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2c16b4138ba22f067373a30dcd25828cbc35de0a848acaeeae423839a568bbc","family_id":"qf-mod28-rvind-qualification","seed":202610286022,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_values\":[9,34],\"y_values\":[-3,41]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0023","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f869ff9b90f2bea199028956682c2eb655d1dae1bdc040709d1be52aff0294b","family_id":"qf-mod28-rvind-qualification","seed":202610286023,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_values\":[10,35],\"y_values\":[-2,42]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0024","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28a85bea920122c903309191fb9d7c5b4b1bdb405d3e2e88e2f78d8bf0f8e46a","family_id":"qf-mod28-rvind-qualification","seed":202610286024,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_values\":[11,36],\"y_values\":[-1,43]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0025","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf3d4545cbafcaab2a5e8046c79ce20b5b7c4561cd3cbba2c4a083b6d0809626","family_id":"qf-mod28-rvind-qualification","seed":202610286025,"semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_values\":[12,37],\"y_values\":[0,44]}"},"fields":{"title":"Every Joint Grid Entry Must Factorize","stem":"Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (x_values,y_values,x_marginal,y_marginal,all [x,y,joint_mass,product] failures in original row-major order,independent?).","answer":{"kind":"expression","value":"([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], Yes)","explanation":"Aggregate both marginals from the same full table and test every value pair, including cells of zero mass. One equal pair or the marginal tables alone cannot certify independence. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], Yes)."},"skills":["Every Joint Grid Entry Must Factorize"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0026","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8d4cda7ff506eecf2f753cfe3b1febdfd6e20af84225dff975d23f855edc87a","family_id":"qf-mod28-rvind-marginal-product","seed":202610286026,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x\":-12,\"x_values\":[-12,13],\"y\":20,\"y_values\":[-3,20]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-12, y=20 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-12, 20, 6/49, 6/7, 1/7, 6/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-12, 20, 6/49, 6/7, 1/7, 6/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0027","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"766e151897d56c142a9cb49b63337f6a4467bc7471c3ce1f121b7b4ad84edf87","family_id":"qf-mod28-rvind-marginal-product","seed":202610286027,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x\":14,\"x_values\":[-11,14],\"y\":-2,\"y_values\":[-2,21]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=14, y=-2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(14, -2, 12/49, 2/7, 6/7, 12/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (14, -2, 12/49, 2/7, 6/7, 12/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0028","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85bf70c97b3b8f07d16819d9e248446c1f68a090d1630a653230f1e8cb34628e","family_id":"qf-mod28-rvind-marginal-product","seed":202610286028,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x\":-10,\"x_values\":[-10,15],\"y\":22,\"y_values\":[-1,22]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-10, y=22 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-10, 22, 4/49, 4/7, 1/7, 4/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-10, 22, 4/49, 4/7, 1/7, 4/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0029","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50f4f63a9a697c876f22da187bcbb30a5da3a809a0292d3195a3c8579b13eb46","family_id":"qf-mod28-rvind-marginal-product","seed":202610286029,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x\":16,\"x_values\":[-9,16],\"y\":0,\"y_values\":[0,23]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=16, y=0 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(16, 0, 24/49, 4/7, 6/7, 24/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (16, 0, 24/49, 4/7, 6/7, 24/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0030","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b735aa043300ec8945067da2592c4aea89060bca70b72de6c8e032f3c35f4345","family_id":"qf-mod28-rvind-marginal-product","seed":202610286030,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x\":-8,\"x_values\":[-8,17],\"y\":24,\"y_values\":[1,24]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-8, y=24 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-8, 24, 2/49, 2/7, 1/7, 2/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-8, 24, 2/49, 2/7, 1/7, 2/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0031","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"065895db00b37cf7cabe1a7f4d259e5c47f1f800101f5a57112d85a851b83b2d","family_id":"qf-mod28-rvind-marginal-product","seed":202610286031,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x\":18,\"x_values\":[-7,18],\"y\":2,\"y_values\":[2,25]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=18, y=2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(18, 2, 5/49, 1/7, 5/7, 5/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (18, 2, 5/49, 1/7, 5/7, 5/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0032","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4423e4daff2a5239eeb2541c8812c87c07fc84c1ee1871be56e439d936617f19","family_id":"qf-mod28-rvind-marginal-product","seed":202610286032,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x\":-6,\"x_values\":[-6,19],\"y\":26,\"y_values\":[3,26]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-6, y=26 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-6, 26, 10/49, 5/7, 2/7, 10/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-6, 26, 10/49, 5/7, 2/7, 10/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0033","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fb5ed3760e171b84b84f98faca5a2b2af7fb9f5fce7aa3ced483f290df566d3","family_id":"qf-mod28-rvind-marginal-product","seed":202610286033,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x\":20,\"x_values\":[-5,20],\"y\":-3,\"y_values\":[-3,27]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=20, y=-3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(20, -3, 15/49, 3/7, 5/7, 15/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (20, -3, 15/49, 3/7, 5/7, 15/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0034","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fed2d1b01929534a907d8b2df5cbc884079e97923102fc384c886b5f58e489f","family_id":"qf-mod28-rvind-marginal-product","seed":202610286034,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x\":-4,\"x_values\":[-4,21],\"y\":28,\"y_values\":[-2,28]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-4, y=28 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-4, 28, 6/49, 3/7, 2/7, 6/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-4, 28, 6/49, 3/7, 2/7, 6/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0035","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd3a5ab0616fa1fb42bf9f4e794930dbfff6172c111e55d02456181151e8ec78","family_id":"qf-mod28-rvind-marginal-product","seed":202610286035,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x\":22,\"x_values\":[-3,22],\"y\":-1,\"y_values\":[-1,29]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=22, y=-1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(22, -1, 25/49, 5/7, 5/7, 25/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (22, -1, 25/49, 5/7, 5/7, 25/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0036","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6949090973dc6934b24d4e2111537c3aeddce8e3ce27d227712a7591492c2a7b","family_id":"qf-mod28-rvind-marginal-product","seed":202610286036,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x\":-2,\"x_values\":[-2,23],\"y\":30,\"y_values\":[0,30]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=-2, y=30 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(-2, 30, 18/49, 6/7, 3/7, 18/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (-2, 30, 18/49, 6/7, 3/7, 18/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0037","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f04026c7855d82c75163313431979e95d3d89cf597d911246b2bc9db23d245","family_id":"qf-mod28-rvind-marginal-product","seed":202610286037,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x\":24,\"x_values\":[-1,24],\"y\":1,\"y_values\":[1,31]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=24, y=1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(24, 1, 8/49, 2/7, 4/7, 8/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (24, 1, 8/49, 2/7, 4/7, 8/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0038","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"337bf0acfc41d6a4298f39d59f4ba43986be2719bcdc4597173bb35ba21304e4","family_id":"qf-mod28-rvind-marginal-product","seed":202610286038,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x\":0,\"x_values\":[0,25],\"y\":32,\"y_values\":[2,32]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=0, y=32 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(0, 32, 12/49, 4/7, 3/7, 12/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 32, 12/49, 4/7, 3/7, 12/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0039","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a11cd7593bffacf6ab5b55c1acc820303fa55a569996761c0f2f7deabd36c3b3","family_id":"qf-mod28-rvind-marginal-product","seed":202610286039,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x\":26,\"x_values\":[1,26],\"y\":3,\"y_values\":[3,33]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=26, y=3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(26, 3, 16/49, 4/7, 4/7, 16/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (26, 3, 16/49, 4/7, 4/7, 16/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0040","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd24c3b38192ad874dbc8124feb006c9c5f01b4b3a3379a568f4d1b2ea160805","family_id":"qf-mod28-rvind-marginal-product","seed":202610286040,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x\":2,\"x_values\":[2,27],\"y\":34,\"y_values\":[-3,34]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=2, y=34 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(2, 34, 6/49, 2/7, 3/7, 6/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 34, 6/49, 2/7, 3/7, 6/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0041","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"648fdfa0c8da811763c11575bcfb46c536924f6a7fd3abec9b8374acace236cf","family_id":"qf-mod28-rvind-marginal-product","seed":202610286041,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x\":28,\"x_values\":[3,28],\"y\":-2,\"y_values\":[-2,35]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=28, y=-2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(28, -2, 3/49, 1/7, 3/7, 3/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (28, -2, 3/49, 1/7, 3/7, 3/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0042","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3c5dfc43412f2c90afc87490a963626487afe996283fa8fbaa3b79069b36a71","family_id":"qf-mod28-rvind-marginal-product","seed":202610286042,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x\":4,\"x_values\":[4,29],\"y\":36,\"y_values\":[-1,36]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=4, y=36 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(4, 36, 20/49, 5/7, 4/7, 20/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 36, 20/49, 5/7, 4/7, 20/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0043","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"470f86ec9ca215d26dc3a379a9796fb81763c3d02e6409f559fb71911e176ee2","family_id":"qf-mod28-rvind-marginal-product","seed":202610286043,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x\":30,\"x_values\":[5,30],\"y\":0,\"y_values\":[0,37]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=30, y=0 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(30, 0, 9/49, 3/7, 3/7, 9/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (30, 0, 9/49, 3/7, 3/7, 9/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0044","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18537ea66919e85c986d51a3ff7ebcebd5bf3f92a07b166200fa46cfea964925","family_id":"qf-mod28-rvind-marginal-product","seed":202610286044,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x\":6,\"x_values\":[6,31],\"y\":38,\"y_values\":[1,38]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=6, y=38 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(6, 38, 12/49, 3/7, 4/7, 12/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 38, 12/49, 3/7, 4/7, 12/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0045","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fe1a56e1c51be361da93e6e67d6b3e65ee1623aeec1c946bcee67026a2d7c22","family_id":"qf-mod28-rvind-marginal-product","seed":202610286045,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x\":32,\"x_values\":[7,32],\"y\":2,\"y_values\":[2,39]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=32, y=2 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(32, 2, 15/49, 5/7, 3/7, 15/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (32, 2, 15/49, 5/7, 3/7, 15/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0046","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc897870e46a1b1a5974c1b1ed93a73e3c497212485eb0a4365edcdfb955130b","family_id":"qf-mod28-rvind-marginal-product","seed":202610286046,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x\":8,\"x_values\":[8,33],\"y\":40,\"y_values\":[3,40]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=8, y=40 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(8, 40, 30/49, 6/7, 5/7, 30/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (8, 40, 30/49, 6/7, 5/7, 30/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0047","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4416dc4a1aae6c8f10967dc408a3410c6c42daac9881a4203cf2f2263e07d044","family_id":"qf-mod28-rvind-marginal-product","seed":202610286047,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x\":34,\"x_values\":[9,34],\"y\":-3,\"y_values\":[-3,41]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=34, y=-3 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(34, -3, 4/49, 2/7, 2/7, 4/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (34, -3, 4/49, 2/7, 2/7, 4/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0048","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c669c91144b846b21271c779d53ab816b425607c0b6b1eb843ce720e71de756","family_id":"qf-mod28-rvind-marginal-product","seed":202610286048,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x\":10,\"x_values\":[10,35],\"y\":42,\"y_values\":[-2,42]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=10, y=42 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(10, 42, 20/49, 4/7, 5/7, 20/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (10, 42, 20/49, 4/7, 5/7, 20/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0049","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d701f9093bb957dbfa1bc2e96a1d96b5c68fa2dcadbddf577329ed80284b3a9","family_id":"qf-mod28-rvind-marginal-product","seed":202610286049,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x\":36,\"x_values\":[11,36],\"y\":-1,\"y_values\":[-1,43]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=36, y=-1 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(36, -1, 8/49, 4/7, 2/7, 8/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (36, -1, 8/49, 4/7, 2/7, 8/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0050","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6622630e04209b8dc594c9fcac96395c8730b039d8cc239b0a5bb61a8362403","family_id":"qf-mod28-rvind-marginal-product","seed":202610286050,"semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x\":12,\"x_values\":[12,37],\"y\":44,\"y_values\":[0,44]}"},"fields":{"title":"One Cell Equality Is Not Whole-Law Independence","stem":"Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Selected value labels x=12, y=44 must occur in their respective grids. Give (x,y,joint_mass,x_mass,y_mass,product,equal?).","answer":{"kind":"expression","value":"(12, 44, 10/49, 2/7, 5/7, 10/49, Yes)","explanation":"Look up by the supplied value labels, not by numerically sorting the grids or treating values as indices. This flag covers only the selected cell. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, 44, 10/49, 2/7, 5/7, 10/49, Yes)."},"skills":["One Cell Equality Is Not Whole-Law Independence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0051","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"277c54d1353d0bb1a04e735c14f135ae04756c3bbb603a45acf9673f2b79937c","family_id":"qf-mod28-rvind-event-preimage","seed":202610286051,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x_event\":[-12],\"x_values\":[-12,13],\"y_event\":[20],\"y_values\":[-3,20]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-12], y_event=[20] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-12], [20], 6/7, 1/7, 6/49, 6/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-12], [20], 6/7, 1/7, 6/49, 6/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0052","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa0d6320c0601c8467f93ab22a4884cea05a2c5b3f6f4420d56e5721b3af31a5","family_id":"qf-mod28-rvind-event-preimage","seed":202610286052,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_event\":[14],\"x_values\":[-11,14],\"y_event\":[-2],\"y_values\":[-2,21]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[14], y_event=[-2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([14], [-2], 2/7, 6/7, 12/49, 12/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([14], [-2], 2/7, 6/7, 12/49, 12/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0053","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd85f1e0b61da798e0bd6ae61fb7d1eccc7806e64d9bce7e5ae782049a1c5988","family_id":"qf-mod28-rvind-event-preimage","seed":202610286053,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_event\":[-10],\"x_values\":[-10,15],\"y_event\":[22],\"y_values\":[-1,22]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-10], y_event=[22] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-10], [22], 4/7, 1/7, 4/49, 4/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-10], [22], 4/7, 1/7, 4/49, 4/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0054","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dd8ed1c314dd8c13b782c998c5d5a9675b76fd1dd87d88dddab529ac99b3f33","family_id":"qf-mod28-rvind-event-preimage","seed":202610286054,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_event\":[16],\"x_values\":[-9,16],\"y_event\":[0],\"y_values\":[0,23]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[16], y_event=[0] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([16], [0], 4/7, 6/7, 24/49, 24/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([16], [0], 4/7, 6/7, 24/49, 24/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0055","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1caaa17562d3b1efa199bf2fc804b326887238eafdbf83645ba8ccf157a5c808","family_id":"qf-mod28-rvind-event-preimage","seed":202610286055,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_event\":[-8],\"x_values\":[-8,17],\"y_event\":[24],\"y_values\":[1,24]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-8], y_event=[24] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-8], [24], 2/7, 1/7, 2/49, 2/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-8], [24], 2/7, 1/7, 2/49, 2/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0056","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82f594d3c7cb998248e14711a8ab1483ab3120d3bd698588ce3a3b5efc4ff88f","family_id":"qf-mod28-rvind-event-preimage","seed":202610286056,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x_event\":[18],\"x_values\":[-7,18],\"y_event\":[2],\"y_values\":[2,25]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[18], y_event=[2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([18], [2], 1/7, 5/7, 5/49, 5/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([18], [2], 1/7, 5/7, 5/49, 5/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0057","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48d4ea68f5ee9e48180de59722b4f832c28da81da449144e5278a592cb8738e7","family_id":"qf-mod28-rvind-event-preimage","seed":202610286057,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_event\":[-6],\"x_values\":[-6,19],\"y_event\":[26],\"y_values\":[3,26]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-6], y_event=[26] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-6], [26], 5/7, 2/7, 10/49, 10/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-6], [26], 5/7, 2/7, 10/49, 10/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0058","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"213309b30d9a1851d6e94658488c387fe641069bff71f93714b844c343fd1f8d","family_id":"qf-mod28-rvind-event-preimage","seed":202610286058,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_event\":[20],\"x_values\":[-5,20],\"y_event\":[-3],\"y_values\":[-3,27]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[20], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([20], [-3], 3/7, 5/7, 15/49, 15/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([20], [-3], 3/7, 5/7, 15/49, 15/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0059","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"570e42ec17adc1bf978ba7e2961a95b044055202d7772be558ba0d73ae01574d","family_id":"qf-mod28-rvind-event-preimage","seed":202610286059,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_event\":[-4],\"x_values\":[-4,21],\"y_event\":[28],\"y_values\":[-2,28]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-4], y_event=[28] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-4], [28], 3/7, 2/7, 6/49, 6/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-4], [28], 3/7, 2/7, 6/49, 6/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0060","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af98bc1b0dc44cd6f769520aafee0db6e39be660e3f2c5d721662aa6bd076303","family_id":"qf-mod28-rvind-event-preimage","seed":202610286060,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_event\":[22],\"x_values\":[-3,22],\"y_event\":[-1],\"y_values\":[-1,29]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[22], y_event=[-1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([22], [-1], 5/7, 5/7, 25/49, 25/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([22], [-1], 5/7, 5/7, 25/49, 25/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0061","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4477655b033b711409f206a00fd66d8c5ac85eae62e7d622a88e2f2052453643","family_id":"qf-mod28-rvind-event-preimage","seed":202610286061,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x_event\":[-2],\"x_values\":[-2,23],\"y_event\":[30],\"y_values\":[0,30]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[-2], y_event=[30] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([-2], [30], 6/7, 3/7, 18/49, 18/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([-2], [30], 6/7, 3/7, 18/49, 18/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0062","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eedce3bdf374df0a96581dd52b5afe79b476353860c9c022705e8936c6c194d","family_id":"qf-mod28-rvind-event-preimage","seed":202610286062,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_event\":[24],\"x_values\":[-1,24],\"y_event\":[1],\"y_values\":[1,31]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[24], y_event=[1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([24], [1], 2/7, 4/7, 8/49, 8/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([24], [1], 2/7, 4/7, 8/49, 8/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0063","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c07a64ac69f0b54416eb315d925d5d37b35b9e41e1cb46d0a49be0edaf13caa2","family_id":"qf-mod28-rvind-event-preimage","seed":202610286063,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_event\":[0],\"x_values\":[0,25],\"y_event\":[32],\"y_values\":[2,32]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[0], y_event=[32] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([0], [32], 4/7, 3/7, 12/49, 12/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0], [32], 4/7, 3/7, 12/49, 12/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0064","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d1e8c182a765330221c7f75c9264e27df808af7bf9f2d2ae41e0da26468a715","family_id":"qf-mod28-rvind-event-preimage","seed":202610286064,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_event\":[26],\"x_values\":[1,26],\"y_event\":[3],\"y_values\":[3,33]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[26], y_event=[3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([26], [3], 4/7, 4/7, 16/49, 16/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([26], [3], 4/7, 4/7, 16/49, 16/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0065","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e6abc88b1452f079933527bd7cbd6c4f0de6bf5d41dc798df9eabea084ed83","family_id":"qf-mod28-rvind-event-preimage","seed":202610286065,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_event\":[2],\"x_values\":[2,27],\"y_event\":[34],\"y_values\":[-3,34]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[2], y_event=[34] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([2], [34], 2/7, 3/7, 6/49, 6/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([2], [34], 2/7, 3/7, 6/49, 6/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0066","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9a99da7b1e4b4e997e310b083626f35760602335c18e89fd5cc177b34ee35d3","family_id":"qf-mod28-rvind-event-preimage","seed":202610286066,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x_event\":[28],\"x_values\":[3,28],\"y_event\":[-2],\"y_values\":[-2,35]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[28], y_event=[-2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([28], [-2], 1/7, 3/7, 3/49, 3/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([28], [-2], 1/7, 3/7, 3/49, 3/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0067","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d595de9fef3ef808397c96ef27325b055e6ae6db55e8845757b348c82522779d","family_id":"qf-mod28-rvind-event-preimage","seed":202610286067,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_event\":[4],\"x_values\":[4,29],\"y_event\":[36],\"y_values\":[-1,36]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[4], y_event=[36] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([4], [36], 5/7, 4/7, 20/49, 20/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([4], [36], 5/7, 4/7, 20/49, 20/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0068","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff7c8ce1a6a090b67ef52bc36d7b4748e7097d30366e60fa2f7838a654d75f8c","family_id":"qf-mod28-rvind-event-preimage","seed":202610286068,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_event\":[30],\"x_values\":[5,30],\"y_event\":[0],\"y_values\":[0,37]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[30], y_event=[0] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([30], [0], 3/7, 3/7, 9/49, 9/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([30], [0], 3/7, 3/7, 9/49, 9/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0069","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bacb2ef1794638f87b3450e6f5962192d6189af653365e5c6880c69911d783c5","family_id":"qf-mod28-rvind-event-preimage","seed":202610286069,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_event\":[6],\"x_values\":[6,31],\"y_event\":[38],\"y_values\":[1,38]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[6], y_event=[38] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([6], [38], 3/7, 4/7, 12/49, 12/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([6], [38], 3/7, 4/7, 12/49, 12/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0070","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf0e2d8fd994af67b8b5882200094440f46e3194c7984e49a5282ab0c966307c","family_id":"qf-mod28-rvind-event-preimage","seed":202610286070,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_event\":[32],\"x_values\":[7,32],\"y_event\":[2],\"y_values\":[2,39]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[32], y_event=[2] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([32], [2], 5/7, 3/7, 15/49, 15/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([32], [2], 5/7, 3/7, 15/49, 15/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0071","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2a532948c8108e48f3f1e04c9e1d781e19a6519e50e329a6bb1d9e6399b563","family_id":"qf-mod28-rvind-event-preimage","seed":202610286071,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x_event\":[8],\"x_values\":[8,33],\"y_event\":[40],\"y_values\":[3,40]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[8], y_event=[40] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([8], [40], 6/7, 5/7, 30/49, 30/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([8], [40], 6/7, 5/7, 30/49, 30/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0072","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52144ae39f0abe7a2ec02ea1690c5c8584445ea9dc3ca4154bec7be80b9a3ea0","family_id":"qf-mod28-rvind-event-preimage","seed":202610286072,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"x_values\":[9,34],\"y_event\":[-3],\"y_values\":[-3,41]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[34], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0073","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6ffe5ce76bcc0534f86ceacc26cd343cf1aa0217024401bd7f1554d12d19dbc","family_id":"qf-mod28-rvind-event-preimage","seed":202610286073,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_event\":[10],\"x_values\":[10,35],\"y_event\":[42],\"y_values\":[-2,42]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[10], y_event=[42] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([10], [42], 4/7, 5/7, 20/49, 20/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([10], [42], 4/7, 5/7, 20/49, 20/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0074","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5712c48eba6ec760f596a0d3e3963fe9cf59cd8ab8e8332b6bd53a1f2e494851","family_id":"qf-mod28-rvind-event-preimage","seed":202610286074,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_event\":[36],\"x_values\":[11,36],\"y_event\":[-1],\"y_values\":[-1,43]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[36], y_event=[-1] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([36], [-1], 4/7, 2/7, 8/49, 8/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([36], [-1], 4/7, 2/7, 8/49, 8/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0075","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96959e8e23facb8adc83f1ea0e369a71da3338a4fce7e30eb7915d253882ddb3","family_id":"qf-mod28-rvind-event-preimage","seed":202610286075,"semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_event\":[12],\"x_values\":[12,37],\"y_event\":[44],\"y_values\":[0,44]}"},"fields":{"title":"A Selected Pair of Value Events under the Same Joint Law","stem":"Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[12], y_event=[44] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).","answer":{"kind":"expression","value":"([12], [44], 2/7, 5/7, 10/49, 10/49, Yes)","explanation":"Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([12], [44], 2/7, 5/7, 10/49, 10/49, Yes)."},"skills":["A Selected Pair of Value Events under the Same 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-20261014-advance-20261028-r06-random-variable-independence-0076","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f38831304089e021bd8d4424a62ddca86a41c6a08a4fade7967494170c467f2","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286076,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x_values\":[-12,13],\"y_values\":[-3,20]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-12, 13], y_values=[-3, 20], and aligned row-by-column exact masses=[[36/49, 6/49], [6/49, 1/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, -59/7, 2/7, -118/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -59/7, 2/7, -118/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0077","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f3458f6eb25404292f44a9e456d05a4eb8281ffc18b9e96a2b1c1950e7714b0","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286077,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_values\":[-11,14],\"y_values\":[-2,21]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-11, 14], y_values=[-2, 21], and aligned row-by-column exact masses=[[30/49, 5/49], [12/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, -27/7, 9/7, -243/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -27/7, 9/7, -243/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0078","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d50e3aa1a642b23e8874dd73b154fb542e697835364e1eabb1119b1e571cf025","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286078,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_values\":[-10,15],\"y_values\":[-1,22]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-10, 15], y_values=[-1, 22], and aligned row-by-column exact masses=[[24/49, 4/49], [18/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 5/7, 16/7, 80/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 5/7, 16/7, 80/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0079","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b307b87b49ee605c7d60dce96e0d7f77120d059d605a1d6b670769d8bce07705","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286079,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_values\":[-9,16],\"y_values\":[0,23]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-9, 16], y_values=[0, 23], and aligned row-by-column exact masses=[[18/49, 3/49], [24/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 37/7, 23/7, 851/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 37/7, 23/7, 851/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0080","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f13229e6cd42f824f569e4e2bdd451435385b2ac9c24df475bb7af38d77259d6","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286080,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_values\":[-8,17],\"y_values\":[1,24]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 69/7, 30/7, 2070/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 69/7, 30/7, 2070/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0081","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"998b40bccc1837425ea031db99ee60d8636a193b0bd0cd8ddff156546d6666ef","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286081,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x_values\":[-7,18],\"y_values\":[2,25]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-7, 18], y_values=[2, 25], and aligned row-by-column exact masses=[[30/49, 12/49], [5/49, 2/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, -24/7, 60/7, -1440/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, -24/7, 60/7, -1440/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0082","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10ff7ded81e9784d288bdec7d85ab1c12cd0ba2e2ea09d812ec548ab2f76afb7","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286082,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_values\":[-6,19],\"y_values\":[3,26]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-6, 19], y_values=[3, 26], and aligned row-by-column exact masses=[[25/49, 10/49], [10/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 8/7, 67/7, 536/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 8/7, 67/7, 536/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0083","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d747549ff2544b3f8ff85e4f043d6648aba937fe6069513283725f1673e6fc75","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286083,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-5, 20], y_values=[-3, 27], and aligned row-by-column exact masses=[[20/49, 8/49], [15/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 40/7, 39/7, 1560/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0084","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8594e6bf3a8ea8fdcc3fc8744192f648bb4b35343a4380aa08dcb71711a1457a","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286084,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_values\":[-4,21],\"y_values\":[-2,28]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-4, 21], y_values=[-2, 28], and aligned row-by-column exact masses=[[15/49, 6/49], [20/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 72/7, 46/7, 3312/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 72/7, 46/7, 3312/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0085","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22be1f6a2deb2867022c9fe9ac2556c9d112c18808aca1ee3f93824ff735803d","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286085,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_values\":[-3,22],\"y_values\":[-1,29]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-3, 22], y_values=[-1, 29], and aligned row-by-column exact masses=[[10/49, 4/49], [25/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 104/7, 53/7, 5512/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 104/7, 53/7, 5512/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0086","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5b453f5c83e52bebea8417b4e59cc60a6862e1de93959c98e05bbd3635d98b","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286086,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x_values\":[-2,23],\"y_values\":[0,30]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-2, 23], y_values=[0, 30], and aligned row-by-column exact masses=[[24/49, 18/49], [4/49, 3/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 11/7, 90/7, 990/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 11/7, 90/7, 990/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0087","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b66eb5aa9e912038d935dcd5ce5248ee94390ea7bf01b33095a7c1adbbb26b28","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286087,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_values\":[-1,24],\"y_values\":[1,31]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[-1, 24], y_values=[1, 31], and aligned row-by-column exact masses=[[20/49, 15/49], [8/49, 6/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 43/7, 97/7, 4171/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 43/7, 97/7, 4171/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0088","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc0708b10e935c854a67730d3030263240b3bca83fd002ff6e75c9f100938330","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286088,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_values\":[0,25],\"y_values\":[2,32]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[0, 25], y_values=[2, 32], and aligned row-by-column exact masses=[[16/49, 12/49], [12/49, 9/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 75/7, 104/7, 7800/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 75/7, 104/7, 7800/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0089","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9111c0468c2193045b70b1d281abef9ed21af5a05e357b084213e957af825c61","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286089,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_values\":[1,26],\"y_values\":[3,33]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 107/7, 111/7, 11877/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 107/7, 111/7, 11877/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0090","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f14099d69ec5290531c3003f21cee0092d56121236340a489b881c792577aa","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286090,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_values\":[2,27],\"y_values\":[-3,34]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[2, 27], y_values=[-3, 34], and aligned row-by-column exact masses=[[8/49, 6/49], [20/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 139/7, 90/7, 12510/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 139/7, 90/7, 12510/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0091","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90cf12371ff5f789f277a684d10e5fd2e4036a74022acb899ff8ea07370c24dd","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286091,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x_values\":[3,28],\"y_values\":[-2,35]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[3, 28], y_values=[-2, 35], and aligned row-by-column exact masses=[[18/49, 24/49], [3/49, 4/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 46/7, 134/7, 6164/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 46/7, 134/7, 6164/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0092","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de540e919205cadb207030a2d2fcc2b9114cc7bafd6fd253b5ccd5b37c8c76e","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286092,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_values\":[4,29],\"y_values\":[-1,36]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[4, 29], y_values=[-1, 36], and aligned row-by-column exact masses=[[15/49, 20/49], [6/49, 8/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 78/7, 141/7, 10998/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 78/7, 141/7, 10998/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0093","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"208c2ac3276f9b5cf700b0ba00bbee83b1ddad275ef0f3089f3220eee9adaccb","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286093,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_values\":[5,30],\"y_values\":[0,37]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[5, 30], y_values=[0, 37], and aligned row-by-column exact masses=[[12/49, 16/49], [9/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 110/7, 148/7, 16280/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 110/7, 148/7, 16280/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0094","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"364adf01acea89f90ed129f70ca8d361b8c6a11a273f5e331869fe390932cb62","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286094,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_values\":[6,31],\"y_values\":[1,38]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[6, 31], y_values=[1, 38], and aligned row-by-column exact masses=[[9/49, 12/49], [12/49, 16/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 142/7, 155/7, 22010/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 142/7, 155/7, 22010/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0095","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca2e95237684a2a419ac2a47000d6307024340708b67f6219f465e29a19f502f","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286095,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_values\":[7,32],\"y_values\":[2,39]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[7, 32], y_values=[2, 39], and aligned row-by-column exact masses=[[6/49, 8/49], [15/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 174/7, 162/7, 28188/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 174/7, 162/7, 28188/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0096","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc73683a64cf9d402534aac7209e78a692c0079d01022ba63a4db142d44437ec","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286096,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x_values\":[8,33],\"y_values\":[3,40]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[8, 33], y_values=[3, 40], and aligned row-by-column exact masses=[[12/49, 30/49], [2/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 81/7, 206/7, 16686/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 81/7, 206/7, 16686/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0097","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cac1e7c5c91420007903705e8237d13da41cc53609f7727ea37a902a344c104","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286097,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_values\":[9,34],\"y_values\":[-3,41]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 113/7, 199/7, 22487/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 113/7, 199/7, 22487/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0098","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf33bbfe48b7828eec8b02a35a7c4d5c59130dd2ac2c97c27b05abaa813c63f4","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286098,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_values\":[10,35],\"y_values\":[-2,42]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[10, 35], y_values=[-2, 42], and aligned row-by-column exact masses=[[8/49, 20/49], [6/49, 15/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 145/7, 206/7, 29870/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 145/7, 206/7, 29870/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0099","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0a88a496e3fceaaf72cb75b4e19dfa56b87d5cdbe9602b21b59aa4592f93ee9","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286099,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_values\":[11,36],\"y_values\":[-1,43]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[11, 36], y_values=[-1, 43], and aligned row-by-column exact masses=[[6/49, 15/49], [8/49, 20/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 177/7, 213/7, 37701/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 177/7, 213/7, 37701/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r06-random-variable-independence-0100","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a340d8617a0814f266a8ab29641f632e0c4bd4feddbb2a69c5e4798dfd2102b","family_id":"qf-mod28-rvind-zero-covariance","seed":202610286100,"semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_values\":[12,37],\"y_values\":[0,44]}"},"fields":{"title":"Keep Verified Independence and Zero Covariance Distinct","stem":"Use the complete joint grid with original x_values=[12, 37], y_values=[0, 44], and aligned row-by-column exact masses=[[4/49, 10/49], [10/49, 25/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.","answer":{"kind":"expression","value":"(Yes, 209/7, 220/7, 45980/49, 0, Yes, zero_covariance_converse_not_claimed)","explanation":"Compute every expectation from this joint table. Verified independence supplies the zero-covariance consequence. When independence is unverified or false, a No conjunction flag does not refute the theorem; zero covariance alone never establishes the missing independence premise. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (Yes, 209/7, 220/7, 45980/49, 0, Yes, zero_covariance_converse_not_claimed)."},"skills":["Keep Verified Independence and Zero Covariance Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}