{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0001","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94c8f829d9fa34a312f4cc502eac1da230bb74efe9be81aa301c44568dad9b1f","family_id":"qf-mod47-suff-fibers","seed":202611165001,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[0,1,2,3],\"statistic\":[0,1,0,1],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 0, 1], [[0, [0, 2]], [1, [1, 3]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 0, 1], [[0, [0, 2]], [1, [1, 3]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0002","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e93ba4f8e5aa4fca35788680456779848e71791d7bdf289888e27c2bd014c3b","family_id":"qf-mod47-suff-fibers","seed":202611165002,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[10,11,12,13],\"statistic\":[0,1,0,1],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([10, 11, 12, 13], [0, 1, 0, 1], [[0, [10, 12]], [1, [11, 13]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([10, 11, 12, 13], [0, 1, 0, 1], [[0, [10, 12]], [1, [11, 13]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0003","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a153ca06e349a6cbb12c9f851bd503a5283ed4165c6a6efbb3f1bbf3bdca15e8","family_id":"qf-mod47-suff-fibers","seed":202611165003,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[20,21,22,23],\"statistic\":[0,1,0,1],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([20, 21, 22, 23], [0, 1, 0, 1], [[0, [20, 22]], [1, [21, 23]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([20, 21, 22, 23], [0, 1, 0, 1], [[0, [20, 22]], [1, [21, 23]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0004","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14eb0b856d465f140a54591d52e30fa787760c5e99eab8d52f8b1f020c646b89","family_id":"qf-mod47-suff-fibers","seed":202611165004,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[30,31,32,33],\"statistic\":[0,1,0,1],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([30, 31, 32, 33], [0, 1, 0, 1], [[0, [30, 32]], [1, [31, 33]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([30, 31, 32, 33], [0, 1, 0, 1], [[0, [30, 32]], [1, [31, 33]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0005","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d776b9ec856759bb2d87f2c16da7d13dadd0dd35f1ddbbe420215a6939cfed5a","family_id":"qf-mod47-suff-fibers","seed":202611165005,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[40,41,42,43],\"statistic\":[0,1,0,1],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([40, 41, 42, 43], [0, 1, 0, 1], [[0, [40, 42]], [1, [41, 43]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([40, 41, 42, 43], [0, 1, 0, 1], [[0, [40, 42]], [1, [41, 43]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0006","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257cf51aeb875a850adce7824d4098db86d709a67086b8e971abcd9b1f85eaf0","family_id":"qf-mod47-suff-fibers","seed":202611165006,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([50, 51, 52, 53], [0, 1, 0, 1], [[0, [50, 52]], [1, [51, 53]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([50, 51, 52, 53], [0, 1, 0, 1], [[0, [50, 52]], [1, [51, 53]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0007","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3a74af5117c2ddd61b4ddccb84f317b2cee2444296a481f0adb84d2fae0988a","family_id":"qf-mod47-suff-fibers","seed":202611165007,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[60,61,62,63],\"statistic\":[0,1,0,1],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([60, 61, 62, 63], [0, 1, 0, 1], [[0, [60, 62]], [1, [61, 63]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([60, 61, 62, 63], [0, 1, 0, 1], [[0, [60, 62]], [1, [61, 63]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0008","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8b167e0d62f282e90a9246b6ee79d528ec86bddb7d7297d025fc8adad7f3912","family_id":"qf-mod47-suff-fibers","seed":202611165008,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[70,71,72,73],\"statistic\":[0,1,0,1],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([70, 71, 72, 73], [0, 1, 0, 1], [[0, [70, 72]], [1, [71, 73]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([70, 71, 72, 73], [0, 1, 0, 1], [[0, [70, 72]], [1, [71, 73]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0009","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39d3b40595f7e25fada10b0464933888ee2aff10577a0cab1d0d098ee3b101e9","family_id":"qf-mod47-suff-fibers","seed":202611165009,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[80,81,82,83],\"statistic\":[0,1,0,1],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([80, 81, 82, 83], [0, 1, 0, 1], [[0, [80, 82]], [1, [81, 83]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([80, 81, 82, 83], [0, 1, 0, 1], [[0, [80, 82]], [1, [81, 83]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0010","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c31bb441bc35247c5812ca9714d57def703812073f1f3194f043556276bf1086","family_id":"qf-mod47-suff-fibers","seed":202611165010,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[90,91,92,93],\"statistic\":[0,1,0,1],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([90, 91, 92, 93], [0, 1, 0, 1], [[0, [90, 92]], [1, [91, 93]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([90, 91, 92, 93], [0, 1, 0, 1], [[0, [90, 92]], [1, [91, 93]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0011","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5e3d0373553fd808257847d00ee98f703281371f2a33ad0a5017f37d91a20b5","family_id":"qf-mod47-suff-fibers","seed":202611165011,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[100,101,102,103],\"statistic\":[0,1,0,1],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([100, 101, 102, 103], [0, 1, 0, 1], [[0, [100, 102]], [1, [101, 103]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([100, 101, 102, 103], [0, 1, 0, 1], [[0, [100, 102]], [1, [101, 103]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0012","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ed6c90f56a39508a5e98a4a17566c1d43b8d1157c42ee7f3ec4d976dcdd8bc7","family_id":"qf-mod47-suff-fibers","seed":202611165012,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[110,111,112,113],\"statistic\":[0,1,0,1],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([110, 111, 112, 113], [0, 1, 0, 1], [[0, [110, 112]], [1, [111, 113]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([110, 111, 112, 113], [0, 1, 0, 1], [[0, [110, 112]], [1, [111, 113]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0013","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70c6b4cf945632c73525fea6520de6470325e6a7ce34d2fdc36f7989a73f5f4a","family_id":"qf-mod47-suff-fibers","seed":202611165013,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[120,121,122,123],\"statistic\":[0,1,0,1],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([120, 121, 122, 123], [0, 1, 0, 1], [[0, [120, 122]], [1, [121, 123]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([120, 121, 122, 123], [0, 1, 0, 1], [[0, [120, 122]], [1, [121, 123]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0014","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecc8351772b9b2043ad8b2d414b494dbd7fce1c44930754c52dcd84902d06b49","family_id":"qf-mod47-suff-fibers","seed":202611165014,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[130,131,132,133],\"statistic\":[0,1,0,1],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([130, 131, 132, 133], [0, 1, 0, 1], [[0, [130, 132]], [1, [131, 133]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([130, 131, 132, 133], [0, 1, 0, 1], [[0, [130, 132]], [1, [131, 133]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0015","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36809e4f74bd5f477947a3024b423f5b8e2b740477b3ac9b850e03c6887ae3e1","family_id":"qf-mod47-suff-fibers","seed":202611165015,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[140,141,142,143],\"statistic\":[0,1,0,1],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([140, 141, 142, 143], [0, 1, 0, 1], [[0, [140, 142]], [1, [141, 143]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([140, 141, 142, 143], [0, 1, 0, 1], [[0, [140, 142]], [1, [141, 143]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0016","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b65da7d36db3ae531353b2bbf1cae65d6bcec64d1311f5fb4f5ad879b4901a08","family_id":"qf-mod47-suff-fibers","seed":202611165016,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[150,151,152,153],\"statistic\":[0,1,0,1],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([150, 151, 152, 153], [0, 1, 0, 1], [[0, [150, 152]], [1, [151, 153]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([150, 151, 152, 153], [0, 1, 0, 1], [[0, [150, 152]], [1, [151, 153]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0017","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce1bfe5d74b0b6a4fe9eba2c8bb8d3515b6b018a3ba7223b919221b90c2a80b6","family_id":"qf-mod47-suff-fibers","seed":202611165017,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[160,161,162,163],\"statistic\":[0,1,0,1],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([160, 161, 162, 163], [0, 1, 0, 1], [[0, [160, 162]], [1, [161, 163]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([160, 161, 162, 163], [0, 1, 0, 1], [[0, [160, 162]], [1, [161, 163]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0018","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce284caa2d5a7916adcf7be4cd7c3e0817c7a0224ff4e5c7d6224f192c1088c","family_id":"qf-mod47-suff-fibers","seed":202611165018,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([170, 171, 172, 173], [0, 1, 0, 1], [[0, [170, 172]], [1, [171, 173]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([170, 171, 172, 173], [0, 1, 0, 1], [[0, [170, 172]], [1, [171, 173]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0019","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11708da8b19b978fee360049d66a4839623d884583be4d5bc9225d781af2a45e","family_id":"qf-mod47-suff-fibers","seed":202611165019,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[180,181,182,183],\"statistic\":[0,1,0,1],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([180, 181, 182, 183], [0, 1, 0, 1], [[0, [180, 182]], [1, [181, 183]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([180, 181, 182, 183], [0, 1, 0, 1], [[0, [180, 182]], [1, [181, 183]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0020","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8167833a19e541b8b0f1c95d4288e2e340758075ba29e82589db70b8b2b9894a","family_id":"qf-mod47-suff-fibers","seed":202611165020,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[190,191,192,193],\"statistic\":[0,1,0,1],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([190, 191, 192, 193], [0, 1, 0, 1], [[0, [190, 192]], [1, [191, 193]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([190, 191, 192, 193], [0, 1, 0, 1], [[0, [190, 192]], [1, [191, 193]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0021","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf2dd22839a4fc6086c7b553065b7d4588ebbc3c65d0183f98fc775b566593f4","family_id":"qf-mod47-suff-fibers","seed":202611165021,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[200,201,202,203],\"statistic\":[0,1,0,1],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([200, 201, 202, 203], [0, 1, 0, 1], [[0, [200, 202]], [1, [201, 203]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([200, 201, 202, 203], [0, 1, 0, 1], [[0, [200, 202]], [1, [201, 203]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0022","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb286692433aec4830e395b40b47f159010d8cd18e8f10819164e94ffd4be3bd","family_id":"qf-mod47-suff-fibers","seed":202611165022,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[210,211,212,213],\"statistic\":[0,1,0,1],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([210, 211, 212, 213], [0, 1, 0, 1], [[0, [210, 212]], [1, [211, 213]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([210, 211, 212, 213], [0, 1, 0, 1], [[0, [210, 212]], [1, [211, 213]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0023","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f261fad5e319909d7051fc9b1dbdc2ac14b23eb6932c76fd516941c02184180a","family_id":"qf-mod47-suff-fibers","seed":202611165023,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[220,221,222,223],\"statistic\":[0,1,0,1],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([220, 221, 222, 223], [0, 1, 0, 1], [[0, [220, 222]], [1, [221, 223]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([220, 221, 222, 223], [0, 1, 0, 1], [[0, [220, 222]], [1, [221, 223]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0024","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a1a14f75465ea1b170991136027d953a21afe3951b4b762d72b7d73b29b907","family_id":"qf-mod47-suff-fibers","seed":202611165024,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,1,0,1],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0025","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f39ce56212d12643ce0f86874523633a53de537b2e3136282ab2cdb985700768","family_id":"qf-mod47-suff-fibers","seed":202611165025,"semantic_template":"qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[240,241,242,243],\"statistic\":[0,1,0,1],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"A Total Statistic Partitions the Complete Sample Carrier into Fibers","stem":"Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).","answer":{"kind":"expression","value":"([240, 241, 242, 243], [0, 1, 0, 1], [[0, [240, 242]], [1, [241, 243]]], Yes, total_statistic_fibers_partition_sample_space)","explanation":"Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([240, 241, 242, 243], [0, 1, 0, 1], [[0, [240, 242]], [1, [241, 243]]], Yes, total_statistic_fibers_partition_sample_space)."},"skills":["A Total Statistic Partitions the Complete Sample Carrier into Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0026","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddbda5008ebf2a659e44b59f3cad93074629ef7c7840241418b3849478f1d1ed","family_id":"qf-mod47-suff-conditional-table","seed":202611165026,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[0,1,2,3],\"statistic\":[0,1,0,1],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0027","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e1b78a0615722e7526d0a0b5a64a2aa59bec7ebee2fe9e2a93d93fac57f8200","family_id":"qf-mod47-suff-conditional-table","seed":202611165027,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[10,11,12,13],\"statistic\":[0,1,0,1],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a1, 0, 4/5, [[10, 1/2], [12, 1/2]]], [a1, 1, 1/5, [[11, 1/2], [13, 1/2]]], [b1, 0, 1/5, [[10, 1/2], [12, 1/2]]], [b1, 1, 4/5, [[11, 1/2], [13, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a1, 0, 4/5, [[10, 1/2], [12, 1/2]]], [a1, 1, 1/5, [[11, 1/2], [13, 1/2]]], [b1, 0, 1/5, [[10, 1/2], [12, 1/2]]], [b1, 1, 4/5, [[11, 1/2], [13, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0028","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bbb5e19f2458f5e4055928ab8ff762e868e8bdd87376c0388e29d8e789d46e3","family_id":"qf-mod47-suff-conditional-table","seed":202611165028,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[20,21,22,23],\"statistic\":[0,1,0,1],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0029","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af7fae5dba7e67b2eef367821c2ebbd5afe642247ede559c69e90a86630d682e","family_id":"qf-mod47-suff-conditional-table","seed":202611165029,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[30,31,32,33],\"statistic\":[0,1,0,1],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a3, 0, 4/5, [[30, 1/2], [32, 1/2]]], [a3, 1, 1/5, [[31, 1/2], [33, 1/2]]], [b3, 0, 1/5, [[30, 1/2], [32, 1/2]]], [b3, 1, 4/5, [[31, 1/2], [33, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a3, 0, 4/5, [[30, 1/2], [32, 1/2]]], [a3, 1, 1/5, [[31, 1/2], [33, 1/2]]], [b3, 0, 1/5, [[30, 1/2], [32, 1/2]]], [b3, 1, 4/5, [[31, 1/2], [33, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0030","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1b84a86d3943f42384db5f8efaa3d3b3174d3d523c23038aaa25893b286bfa3","family_id":"qf-mod47-suff-conditional-table","seed":202611165030,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[40,41,42,43],\"statistic\":[0,1,0,1],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a4, 0, 4/5, [[40, 1/2], [42, 1/2]]], [a4, 1, 1/5, [[41, 1/2], [43, 1/2]]], [b4, 0, 1/5, [[40, 1/2], [42, 1/2]]], [b4, 1, 4/5, [[41, 1/2], [43, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a4, 0, 4/5, [[40, 1/2], [42, 1/2]]], [a4, 1, 1/5, [[41, 1/2], [43, 1/2]]], [b4, 0, 1/5, [[40, 1/2], [42, 1/2]]], [b4, 1, 4/5, [[41, 1/2], [43, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0031","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81bed7b17bad05247d1d3a79b85894323ca253a0976bda1cfa7c1c7a7859eb25","family_id":"qf-mod47-suff-conditional-table","seed":202611165031,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a5, 0, 4/5, [[50, 1/2], [52, 1/2]]], [a5, 1, 1/5, [[51, 1/2], [53, 1/2]]], [b5, 0, 1/5, [[50, 1/2], [52, 1/2]]], [b5, 1, 4/5, [[51, 1/2], [53, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a5, 0, 4/5, [[50, 1/2], [52, 1/2]]], [a5, 1, 1/5, [[51, 1/2], [53, 1/2]]], [b5, 0, 1/5, [[50, 1/2], [52, 1/2]]], [b5, 1, 4/5, [[51, 1/2], [53, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0032","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5712cd0c7820e2b2a59464f3ec38b1be93f787607a2f1ead9e8848cb39b62445","family_id":"qf-mod47-suff-conditional-table","seed":202611165032,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[60,61,62,63],\"statistic\":[0,1,0,1],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a6, 0, 4/5, [[60, 1/2], [62, 1/2]]], [a6, 1, 1/5, [[61, 1/2], [63, 1/2]]], [b6, 0, 1/5, [[60, 1/2], [62, 1/2]]], [b6, 1, 4/5, [[61, 1/2], [63, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a6, 0, 4/5, [[60, 1/2], [62, 1/2]]], [a6, 1, 1/5, [[61, 1/2], [63, 1/2]]], [b6, 0, 1/5, [[60, 1/2], [62, 1/2]]], [b6, 1, 4/5, [[61, 1/2], [63, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0033","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9305c912d4587a8d7bb9e3f778ee961c6b14d5b30a11c398245a1137e92150a","family_id":"qf-mod47-suff-conditional-table","seed":202611165033,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[70,71,72,73],\"statistic\":[0,1,0,1],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a7, 0, 4/5, [[70, 1/2], [72, 1/2]]], [a7, 1, 1/5, [[71, 1/2], [73, 1/2]]], [b7, 0, 1/5, [[70, 1/2], [72, 1/2]]], [b7, 1, 4/5, [[71, 1/2], [73, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a7, 0, 4/5, [[70, 1/2], [72, 1/2]]], [a7, 1, 1/5, [[71, 1/2], [73, 1/2]]], [b7, 0, 1/5, [[70, 1/2], [72, 1/2]]], [b7, 1, 4/5, [[71, 1/2], [73, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0034","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"667e2337f87f26607f90280c841dc492903c179e15a1a1fcdf74ce12095c2047","family_id":"qf-mod47-suff-conditional-table","seed":202611165034,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[80,81,82,83],\"statistic\":[0,1,0,1],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a8, 0, 4/5, [[80, 1/2], [82, 1/2]]], [a8, 1, 1/5, [[81, 1/2], [83, 1/2]]], [b8, 0, 1/5, [[80, 1/2], [82, 1/2]]], [b8, 1, 4/5, [[81, 1/2], [83, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a8, 0, 4/5, [[80, 1/2], [82, 1/2]]], [a8, 1, 1/5, [[81, 1/2], [83, 1/2]]], [b8, 0, 1/5, [[80, 1/2], [82, 1/2]]], [b8, 1, 4/5, [[81, 1/2], [83, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0035","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f04e7264db1cdbad44d1f80b75332479a89ae5794843ef17b8959841098f1380","family_id":"qf-mod47-suff-conditional-table","seed":202611165035,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[90,91,92,93],\"statistic\":[0,1,0,1],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a9, 0, 4/5, [[90, 1/2], [92, 1/2]]], [a9, 1, 1/5, [[91, 1/2], [93, 1/2]]], [b9, 0, 1/5, [[90, 1/2], [92, 1/2]]], [b9, 1, 4/5, [[91, 1/2], [93, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a9, 0, 4/5, [[90, 1/2], [92, 1/2]]], [a9, 1, 1/5, [[91, 1/2], [93, 1/2]]], [b9, 0, 1/5, [[90, 1/2], [92, 1/2]]], [b9, 1, 4/5, [[91, 1/2], [93, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0036","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8822f6db19282d28801474c842b71ba29471e29879e9d38cf63de474c6e0001b","family_id":"qf-mod47-suff-conditional-table","seed":202611165036,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[100,101,102,103],\"statistic\":[0,1,0,1],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a10, 0, 4/5, [[100, 1/2], [102, 1/2]]], [a10, 1, 1/5, [[101, 1/2], [103, 1/2]]], [b10, 0, 1/5, [[100, 1/2], [102, 1/2]]], [b10, 1, 4/5, [[101, 1/2], [103, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a10, 0, 4/5, [[100, 1/2], [102, 1/2]]], [a10, 1, 1/5, [[101, 1/2], [103, 1/2]]], [b10, 0, 1/5, [[100, 1/2], [102, 1/2]]], [b10, 1, 4/5, [[101, 1/2], [103, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0037","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1be3ff41e7be1755eaba6ca6e39869ad698c8a3dc7d3676841ef7c67ea73935","family_id":"qf-mod47-suff-conditional-table","seed":202611165037,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[110,111,112,113],\"statistic\":[0,1,0,1],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a11, 0, 4/5, [[110, 1/2], [112, 1/2]]], [a11, 1, 1/5, [[111, 1/2], [113, 1/2]]], [b11, 0, 1/5, [[110, 1/2], [112, 1/2]]], [b11, 1, 4/5, [[111, 1/2], [113, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a11, 0, 4/5, [[110, 1/2], [112, 1/2]]], [a11, 1, 1/5, [[111, 1/2], [113, 1/2]]], [b11, 0, 1/5, [[110, 1/2], [112, 1/2]]], [b11, 1, 4/5, [[111, 1/2], [113, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0038","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ee35541a7bdffcb96f07406c0c5d058f903079bb80febcc05efcabee297b347","family_id":"qf-mod47-suff-conditional-table","seed":202611165038,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[120,121,122,123],\"statistic\":[0,1,0,1],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a12, 0, 4/5, [[120, 1/2], [122, 1/2]]], [a12, 1, 1/5, [[121, 1/2], [123, 1/2]]], [b12, 0, 1/5, [[120, 1/2], [122, 1/2]]], [b12, 1, 4/5, [[121, 1/2], [123, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a12, 0, 4/5, [[120, 1/2], [122, 1/2]]], [a12, 1, 1/5, [[121, 1/2], [123, 1/2]]], [b12, 0, 1/5, [[120, 1/2], [122, 1/2]]], [b12, 1, 4/5, [[121, 1/2], [123, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0039","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51c7279066f8d4003545a2b6e9b2e2e9d8bdb086e7480be85e41382b542dc038","family_id":"qf-mod47-suff-conditional-table","seed":202611165039,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[130,131,132,133],\"statistic\":[0,1,0,1],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a13, 0, 4/5, [[130, 1/2], [132, 1/2]]], [a13, 1, 1/5, [[131, 1/2], [133, 1/2]]], [b13, 0, 1/5, [[130, 1/2], [132, 1/2]]], [b13, 1, 4/5, [[131, 1/2], [133, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a13, 0, 4/5, [[130, 1/2], [132, 1/2]]], [a13, 1, 1/5, [[131, 1/2], [133, 1/2]]], [b13, 0, 1/5, [[130, 1/2], [132, 1/2]]], [b13, 1, 4/5, [[131, 1/2], [133, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0040","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ffafa532e5f2e7cb5fd2f6cedcd992fbfc41281250f4fc1777967c90e68da52","family_id":"qf-mod47-suff-conditional-table","seed":202611165040,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[140,141,142,143],\"statistic\":[0,1,0,1],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a14, 0, 4/5, [[140, 1/2], [142, 1/2]]], [a14, 1, 1/5, [[141, 1/2], [143, 1/2]]], [b14, 0, 1/5, [[140, 1/2], [142, 1/2]]], [b14, 1, 4/5, [[141, 1/2], [143, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a14, 0, 4/5, [[140, 1/2], [142, 1/2]]], [a14, 1, 1/5, [[141, 1/2], [143, 1/2]]], [b14, 0, 1/5, [[140, 1/2], [142, 1/2]]], [b14, 1, 4/5, [[141, 1/2], [143, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0041","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d121278182bc62e1034ff79ba087f4b577a1e6ad88f68826c892a28ac2b78d4","family_id":"qf-mod47-suff-conditional-table","seed":202611165041,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[150,151,152,153],\"statistic\":[0,1,0,1],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a15, 0, 4/5, [[150, 1/2], [152, 1/2]]], [a15, 1, 1/5, [[151, 1/2], [153, 1/2]]], [b15, 0, 1/5, [[150, 1/2], [152, 1/2]]], [b15, 1, 4/5, [[151, 1/2], [153, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a15, 0, 4/5, [[150, 1/2], [152, 1/2]]], [a15, 1, 1/5, [[151, 1/2], [153, 1/2]]], [b15, 0, 1/5, [[150, 1/2], [152, 1/2]]], [b15, 1, 4/5, [[151, 1/2], [153, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0042","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e99f2e2800469bb2379f79431da6c652bce017a0f45e3ee8e15d2aaa36a514","family_id":"qf-mod47-suff-conditional-table","seed":202611165042,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[160,161,162,163],\"statistic\":[0,1,0,1],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a16, 0, 4/5, [[160, 1/2], [162, 1/2]]], [a16, 1, 1/5, [[161, 1/2], [163, 1/2]]], [b16, 0, 1/5, [[160, 1/2], [162, 1/2]]], [b16, 1, 4/5, [[161, 1/2], [163, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a16, 0, 4/5, [[160, 1/2], [162, 1/2]]], [a16, 1, 1/5, [[161, 1/2], [163, 1/2]]], [b16, 0, 1/5, [[160, 1/2], [162, 1/2]]], [b16, 1, 4/5, [[161, 1/2], [163, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0043","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c901ad811b45fb1b5a88be077f53e311aad3d6abc22108c287d5dd70c3d483b3","family_id":"qf-mod47-suff-conditional-table","seed":202611165043,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a17, 0, 4/5, [[170, 1/2], [172, 1/2]]], [a17, 1, 1/5, [[171, 1/2], [173, 1/2]]], [b17, 0, 1/5, [[170, 1/2], [172, 1/2]]], [b17, 1, 4/5, [[171, 1/2], [173, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a17, 0, 4/5, [[170, 1/2], [172, 1/2]]], [a17, 1, 1/5, [[171, 1/2], [173, 1/2]]], [b17, 0, 1/5, [[170, 1/2], [172, 1/2]]], [b17, 1, 4/5, [[171, 1/2], [173, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0044","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3163ea7a60c6080e31233843548769f5a0441f6380c406a556fdd6a7aa7c2a4","family_id":"qf-mod47-suff-conditional-table","seed":202611165044,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[180,181,182,183],\"statistic\":[0,1,0,1],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a18, 0, 4/5, [[180, 1/2], [182, 1/2]]], [a18, 1, 1/5, [[181, 1/2], [183, 1/2]]], [b18, 0, 1/5, [[180, 1/2], [182, 1/2]]], [b18, 1, 4/5, [[181, 1/2], [183, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a18, 0, 4/5, [[180, 1/2], [182, 1/2]]], [a18, 1, 1/5, [[181, 1/2], [183, 1/2]]], [b18, 0, 1/5, [[180, 1/2], [182, 1/2]]], [b18, 1, 4/5, [[181, 1/2], [183, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0045","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef58ce95eb9c0ffc347863d5f9b76f8ff3cb07d9d3baf2f8ee35d748fcc37c4a","family_id":"qf-mod47-suff-conditional-table","seed":202611165045,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[190,191,192,193],\"statistic\":[0,1,0,1],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a19, 0, 4/5, [[190, 1/2], [192, 1/2]]], [a19, 1, 1/5, [[191, 1/2], [193, 1/2]]], [b19, 0, 1/5, [[190, 1/2], [192, 1/2]]], [b19, 1, 4/5, [[191, 1/2], [193, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a19, 0, 4/5, [[190, 1/2], [192, 1/2]]], [a19, 1, 1/5, [[191, 1/2], [193, 1/2]]], [b19, 0, 1/5, [[190, 1/2], [192, 1/2]]], [b19, 1, 4/5, [[191, 1/2], [193, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0046","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f6b86607c9aa9d30ec699e012a6180009c192eb3b44d46243dc530e9f445a92","family_id":"qf-mod47-suff-conditional-table","seed":202611165046,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[200,201,202,203],\"statistic\":[0,1,0,1],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0047","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bd7435ccaa5b98334c4c8f77135ef8e931f60079abf6c075a0fc730246bbdcc","family_id":"qf-mod47-suff-conditional-table","seed":202611165047,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[210,211,212,213],\"statistic\":[0,1,0,1],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a21, 0, 4/5, [[210, 1/2], [212, 1/2]]], [a21, 1, 1/5, [[211, 1/2], [213, 1/2]]], [b21, 0, 1/5, [[210, 1/2], [212, 1/2]]], [b21, 1, 4/5, [[211, 1/2], [213, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a21, 0, 4/5, [[210, 1/2], [212, 1/2]]], [a21, 1, 1/5, [[211, 1/2], [213, 1/2]]], [b21, 0, 1/5, [[210, 1/2], [212, 1/2]]], [b21, 1, 4/5, [[211, 1/2], [213, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0048","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daf9c02bd3631e36ecbb4a5b2a8712109460139f1f41791d29a2c9d3f2be31ba","family_id":"qf-mod47-suff-conditional-table","seed":202611165048,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[220,221,222,223],\"statistic\":[0,1,0,1],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a22, 0, 4/5, [[220, 1/2], [222, 1/2]]], [a22, 1, 1/5, [[221, 1/2], [223, 1/2]]], [b22, 0, 1/5, [[220, 1/2], [222, 1/2]]], [b22, 1, 4/5, [[221, 1/2], [223, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a22, 0, 4/5, [[220, 1/2], [222, 1/2]]], [a22, 1, 1/5, [[221, 1/2], [223, 1/2]]], [b22, 0, 1/5, [[220, 1/2], [222, 1/2]]], [b22, 1, 4/5, [[221, 1/2], [223, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0049","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f69ee906d1952f88fc85b76e264371ffb25d63cbdbd3c9a17199b977a8d125","family_id":"qf-mod47-suff-conditional-table","seed":202611165049,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,1,0,1],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a23, 0, 4/5, [[230, 1/2], [232, 1/2]]], [a23, 1, 1/5, [[231, 1/2], [233, 1/2]]], [b23, 0, 1/5, [[230, 1/2], [232, 1/2]]], [b23, 1, 4/5, [[231, 1/2], [233, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a23, 0, 4/5, [[230, 1/2], [232, 1/2]]], [a23, 1, 1/5, [[231, 1/2], [233, 1/2]]], [b23, 0, 1/5, [[230, 1/2], [232, 1/2]]], [b23, 1, 4/5, [[231, 1/2], [233, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0050","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f365fa49791b59ccc7ec4bc187628e27c357440d97b86f9fe2159c57d5541ebe","family_id":"qf-mod47-suff-conditional-table","seed":202611165050,"semantic_template":"qf-mod47-suff-conditional-table/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[240,241,242,243],\"statistic\":[0,1,0,1],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"Conditional Laws Are Defined Only on Positive-Mass Statistic Cells","stem":"Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).","answer":{"kind":"expression","value":"([[a24, 0, 4/5, [[240, 1/2], [242, 1/2]]], [a24, 1, 1/5, [[241, 1/2], [243, 1/2]]], [b24, 0, 1/5, [[240, 1/2], [242, 1/2]]], [b24, 1, 4/5, [[241, 1/2], [243, 1/2]]]], zero_mass_cells_remain_undefined)","explanation":"For every theta and fiber first sum its mass. On a positive-mass cell divide each aligned sample mass by the fiber mass. For mass0 retain null as the encoding of an undefined conditional law, not a fabricated zero distribution or empty table. The exact result, retaining the requested order, is ([[a24, 0, 4/5, [[240, 1/2], [242, 1/2]]], [a24, 1, 1/5, [[241, 1/2], [243, 1/2]]], [b24, 0, 1/5, [[240, 1/2], [242, 1/2]]], [b24, 1, 4/5, [[241, 1/2], [243, 1/2]]]], zero_mass_cells_remain_undefined)."},"skills":["Conditional Laws Are Defined Only on Positive-Mass Statistic Cells"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0051","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"744b1bfab942599f9b272c06d160ba628d3a5594ad2e6392bc2846375779c3cf","family_id":"qf-mod47-suff-invariance","seed":202611165051,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[0,1,2,3],\"statistic\":[0,1,0,1],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a0, [[0, 1/2], [2, 1/2]]], [b0, [[0, 1/2], [2, 1/2]]]], Yes], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a0, [[0, 1/2], [2, 1/2]]], [b0, [[0, 1/2], [2, 1/2]]]], Yes], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0052","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58aa9a88a5de18dc45d4908f13d1e59c6a32681173fd22fccfc968ac9ad3ffed","family_id":"qf-mod47-suff-invariance","seed":202611165052,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[10,11,12,13],\"statistic\":[0,1,0,1],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0053","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e56205623f52ce37cc6fdca5f0f3be6b38f97cb3f92c7448609f8cbf981587","family_id":"qf-mod47-suff-invariance","seed":202611165053,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[20,21,22,23],\"statistic\":[0,1,0,1],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a2, [[20, 1/2], [22, 1/2]]], [b2, [[20, 1/2], [22, 1/2]]]], Yes], [1, [[a2, [[21, 1/2], [23, 1/2]]], [b2, [[21, 1/2], [23, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a2, [[20, 1/2], [22, 1/2]]], [b2, [[20, 1/2], [22, 1/2]]]], Yes], [1, [[a2, [[21, 1/2], [23, 1/2]]], [b2, [[21, 1/2], [23, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0054","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8286b8eee50892949f23ec99e166164c17dc46ff493cd2fe5b6fc38f833b9b41","family_id":"qf-mod47-suff-invariance","seed":202611165054,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[30,31,32,33],\"statistic\":[0,1,0,1],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0055","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3281b82dfc6af1fc2b6322350de95b6ea8b5a1aea67637159821559b1887b4dc","family_id":"qf-mod47-suff-invariance","seed":202611165055,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[40,41,42,43],\"statistic\":[0,1,0,1],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0056","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2263a390350046495a6849b6ba20733b58697a316373dc406388e923c91a15c3","family_id":"qf-mod47-suff-invariance","seed":202611165056,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a5, [[50, 1/2], [52, 1/2]]], [b5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[a5, [[51, 1/2], [53, 1/2]]], [b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a5, [[50, 1/2], [52, 1/2]]], [b5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[a5, [[51, 1/2], [53, 1/2]]], [b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0057","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f489ab159d6814de7cd964a80539f1a9e765dc7b622871497fce08f7275619a","family_id":"qf-mod47-suff-invariance","seed":202611165057,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[60,61,62,63],\"statistic\":[0,1,0,1],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a6, [[60, 1/2], [62, 1/2]]], [b6, [[60, 1/2], [62, 1/2]]]], Yes], [1, [[a6, [[61, 1/2], [63, 1/2]]], [b6, [[61, 1/2], [63, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a6, [[60, 1/2], [62, 1/2]]], [b6, [[60, 1/2], [62, 1/2]]]], Yes], [1, [[a6, [[61, 1/2], [63, 1/2]]], [b6, [[61, 1/2], [63, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0058","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44b675791faf2e48beba8614a3e31a522f4496980c0b29650533ec928ed85a8f","family_id":"qf-mod47-suff-invariance","seed":202611165058,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[70,71,72,73],\"statistic\":[0,1,0,1],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a7, [[70, 1/2], [72, 1/2]]], [b7, [[70, 1/2], [72, 1/2]]]], Yes], [1, [[a7, [[71, 1/2], [73, 1/2]]], [b7, [[71, 1/2], [73, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a7, [[70, 1/2], [72, 1/2]]], [b7, [[70, 1/2], [72, 1/2]]]], Yes], [1, [[a7, [[71, 1/2], [73, 1/2]]], [b7, [[71, 1/2], [73, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0059","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec0e4715d8f8b89d221b024ce04aa1ee8e1ea9322ad14c09fb845a461917bb4b","family_id":"qf-mod47-suff-invariance","seed":202611165059,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[80,81,82,83],\"statistic\":[0,1,0,1],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a8, [[80, 1/2], [82, 1/2]]], [b8, [[80, 1/2], [82, 1/2]]]], Yes], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a8, [[80, 1/2], [82, 1/2]]], [b8, [[80, 1/2], [82, 1/2]]]], Yes], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0060","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3cfd69ac3f4183024f010f16a28123edb8cbd8a348350eb8f84411e7963b3e8","family_id":"qf-mod47-suff-invariance","seed":202611165060,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[90,91,92,93],\"statistic\":[0,1,0,1],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a9, [[90, 1/2], [92, 1/2]]], [b9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[a9, [[91, 1/2], [93, 1/2]]], [b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a9, [[90, 1/2], [92, 1/2]]], [b9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[a9, [[91, 1/2], [93, 1/2]]], [b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0061","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6f9261465c69ab3ff98a7ca92ca48d8b74d4546529fae9885231b3d041f339d","family_id":"qf-mod47-suff-invariance","seed":202611165061,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[100,101,102,103],\"statistic\":[0,1,0,1],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a10, [[100, 1/2], [102, 1/2]]], [b10, [[100, 1/2], [102, 1/2]]]], Yes], [1, [[a10, [[101, 1/2], [103, 1/2]]], [b10, [[101, 1/2], [103, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a10, [[100, 1/2], [102, 1/2]]], [b10, [[100, 1/2], [102, 1/2]]]], Yes], [1, [[a10, [[101, 1/2], [103, 1/2]]], [b10, [[101, 1/2], [103, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0062","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"517e9fe6526f2bf6ec1ae349a3f94f3fb6f2eb704ceac288d4f729e994af3522","family_id":"qf-mod47-suff-invariance","seed":202611165062,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[110,111,112,113],\"statistic\":[0,1,0,1],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a11, [[110, 1/2], [112, 1/2]]], [b11, [[110, 1/2], [112, 1/2]]]], Yes], [1, [[a11, [[111, 1/2], [113, 1/2]]], [b11, [[111, 1/2], [113, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a11, [[110, 1/2], [112, 1/2]]], [b11, [[110, 1/2], [112, 1/2]]]], Yes], [1, [[a11, [[111, 1/2], [113, 1/2]]], [b11, [[111, 1/2], [113, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0063","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ae8140f4dfd9382a3ee8fa6ce5757c29b3c6039709dca3c3be35fadfb32c27","family_id":"qf-mod47-suff-invariance","seed":202611165063,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[120,121,122,123],\"statistic\":[0,1,0,1],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a12, [[120, 1/2], [122, 1/2]]], [b12, [[120, 1/2], [122, 1/2]]]], Yes], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a12, [[120, 1/2], [122, 1/2]]], [b12, [[120, 1/2], [122, 1/2]]]], Yes], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0064","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e34baea56c459377080ca6804ae629b775460308efa18cb92a5c6eed9b4d1663","family_id":"qf-mod47-suff-invariance","seed":202611165064,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[130,131,132,133],\"statistic\":[0,1,0,1],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a13, [[130, 1/2], [132, 1/2]]], [b13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[a13, [[131, 1/2], [133, 1/2]]], [b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a13, [[130, 1/2], [132, 1/2]]], [b13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[a13, [[131, 1/2], [133, 1/2]]], [b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0065","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0513e9617052a24201eea436ae94ce8cbdab399091c9c47a98e0e17143ca77","family_id":"qf-mod47-suff-invariance","seed":202611165065,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[140,141,142,143],\"statistic\":[0,1,0,1],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a14, [[140, 1/2], [142, 1/2]]], [b14, [[140, 1/2], [142, 1/2]]]], Yes], [1, [[a14, [[141, 1/2], [143, 1/2]]], [b14, [[141, 1/2], [143, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a14, [[140, 1/2], [142, 1/2]]], [b14, [[140, 1/2], [142, 1/2]]]], Yes], [1, [[a14, [[141, 1/2], [143, 1/2]]], [b14, [[141, 1/2], [143, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0066","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c027b6655079a1566a72bc7335111c406fecdadff0607b337a4968756e79fc0e","family_id":"qf-mod47-suff-invariance","seed":202611165066,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[150,151,152,153],\"statistic\":[0,1,0,1],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a15, [[150, 1/2], [152, 1/2]]], [b15, [[150, 1/2], [152, 1/2]]]], Yes], [1, [[a15, [[151, 1/2], [153, 1/2]]], [b15, [[151, 1/2], [153, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a15, [[150, 1/2], [152, 1/2]]], [b15, [[150, 1/2], [152, 1/2]]]], Yes], [1, [[a15, [[151, 1/2], [153, 1/2]]], [b15, [[151, 1/2], [153, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0067","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfe7efa7f4a8a04915ccabd9c8625455c007eb49be98ffa96d31df7c07c639be","family_id":"qf-mod47-suff-invariance","seed":202611165067,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[160,161,162,163],\"statistic\":[0,1,0,1],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a16, [[160, 1/2], [162, 1/2]]], [b16, [[160, 1/2], [162, 1/2]]]], Yes], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a16, [[160, 1/2], [162, 1/2]]], [b16, [[160, 1/2], [162, 1/2]]]], Yes], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0068","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74a865e13553257b2d210a5c1c2a7cf323ef76daace35920f27536364aac0e09","family_id":"qf-mod47-suff-invariance","seed":202611165068,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0069","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f403cd5f29d962f55cbb3da1f44f4a72bb3504005387160a17fde592cc97d2b6","family_id":"qf-mod47-suff-invariance","seed":202611165069,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[180,181,182,183],\"statistic\":[0,1,0,1],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a18, [[180, 1/2], [182, 1/2]]], [b18, [[180, 1/2], [182, 1/2]]]], Yes], [1, [[a18, [[181, 1/2], [183, 1/2]]], [b18, [[181, 1/2], [183, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a18, [[180, 1/2], [182, 1/2]]], [b18, [[180, 1/2], [182, 1/2]]]], Yes], [1, [[a18, [[181, 1/2], [183, 1/2]]], [b18, [[181, 1/2], [183, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0070","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486d8b411b8a45088a95bb46b9e45f633d4f60ddd0bd3751e699689a6fdb28c9","family_id":"qf-mod47-suff-invariance","seed":202611165070,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[190,191,192,193],\"statistic\":[0,1,0,1],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a19, [[190, 1/2], [192, 1/2]]], [b19, [[190, 1/2], [192, 1/2]]]], Yes], [1, [[a19, [[191, 1/2], [193, 1/2]]], [b19, [[191, 1/2], [193, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a19, [[190, 1/2], [192, 1/2]]], [b19, [[190, 1/2], [192, 1/2]]]], Yes], [1, [[a19, [[191, 1/2], [193, 1/2]]], [b19, [[191, 1/2], [193, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0071","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c01c01d4389e45db06df0d0d6144c645dd779e2b1ec61a3f0c740839ee48ab","family_id":"qf-mod47-suff-invariance","seed":202611165071,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[200,201,202,203],\"statistic\":[0,1,0,1],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a20, [[200, 1/2], [202, 1/2]]], [b20, [[200, 1/2], [202, 1/2]]]], Yes], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a20, [[200, 1/2], [202, 1/2]]], [b20, [[200, 1/2], [202, 1/2]]]], Yes], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0072","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b358c3fb5206856bb9077a1cc067054d3c523990d9e833854f7aa96daa95fd22","family_id":"qf-mod47-suff-invariance","seed":202611165072,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[210,211,212,213],\"statistic\":[0,1,0,1],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0073","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"678b9d584acae8044bf176542b09ea1673a11ad047de4b4892cd45ba2212f710","family_id":"qf-mod47-suff-invariance","seed":202611165073,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[220,221,222,223],\"statistic\":[0,1,0,1],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a22, [[220, 1/2], [222, 1/2]]], [b22, [[220, 1/2], [222, 1/2]]]], Yes], [1, [[a22, [[221, 1/2], [223, 1/2]]], [b22, [[221, 1/2], [223, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a22, [[220, 1/2], [222, 1/2]]], [b22, [[220, 1/2], [222, 1/2]]]], Yes], [1, [[a22, [[221, 1/2], [223, 1/2]]], [b22, [[221, 1/2], [223, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0074","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbdc2e88a7fe8a584181ae11a6572fcb21b444288867929b53264f94f44ad97f","family_id":"qf-mod47-suff-invariance","seed":202611165074,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,1,0,1],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a23, [[230, 1/2], [232, 1/2]]], [b23, [[230, 1/2], [232, 1/2]]]], Yes], [1, [[a23, [[231, 1/2], [233, 1/2]]], [b23, [[231, 1/2], [233, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a23, [[230, 1/2], [232, 1/2]]], [b23, [[230, 1/2], [232, 1/2]]]], Yes], [1, [[a23, [[231, 1/2], [233, 1/2]]], [b23, [[231, 1/2], [233, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0075","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5693e9a9bbbd60d7847683b24b8e3d47b1297a2010f8fcb0509d5437eb7b2693","family_id":"qf-mod47-suff-invariance","seed":202611165075,"semantic_template":"qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[240,241,242,243],\"statistic\":[0,1,0,1],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber","stem":"Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).","answer":{"kind":"expression","value":"([[0, [[a24, [[240, 1/2], [242, 1/2]]], [b24, [[240, 1/2], [242, 1/2]]]], Yes], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)","explanation":"For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a24, [[240, 1/2], [242, 1/2]]], [b24, [[240, 1/2], [242, 1/2]]]], Yes], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)."},"skills":["Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0076","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa9a3436a8c4a0f8bf9fe18dfc5aebf96e9cf70aebcfe909454fdbadb002c840","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165076,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[0,1,2,3],\"statistic\":[0,1,0,1],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], No, [], [0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], No, [], [0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0077","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"947c5c8f9823ab9dd62ad277ff4382d7eddcc8857abc3d4ab6e452d2e4f21680","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165077,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[10,11,12,13],\"statistic\":[0,1,0,1],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, [[a1, 1, 0, null], [b1, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[b1, [[11, 1/2], [13, 1/2]]]], Yes]], Yes, [[a1, 1, 0, null], [b1, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0078","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f01ceab6c38315031407ff6f4805498dce3251988f00c36967d32ebf8d2dbb3","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165078,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[20,21,22,23],\"statistic\":[0,1,2,3],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a2, [[20, 1]]], [b2, [[20, 1]]]], Yes], [1, [[a2, [[21, 1]]], [b2, [[21, 1]]]], Yes], [2, [[a2, [[22, 1]]], [b2, [[22, 1]]]], Yes], [3, [[a2, [[23, 1]]], [b2, [[23, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a2, [[20, 1]]], [b2, [[20, 1]]]], Yes], [1, [[a2, [[21, 1]]], [b2, [[21, 1]]]], Yes], [2, [[a2, [[22, 1]]], [b2, [[22, 1]]]], Yes], [3, [[a2, [[23, 1]]], [b2, [[23, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0079","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ea922ee1159ca683c594f78237956f390ea940a8f35d7b0da694b89f1c1391b","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165079,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[30,31,32,33],\"statistic\":[0,0,0,0],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a3, [[30, 2/5], [31, 1/10], [32, 2/5], [33, 1/10]]], [b3, [[30, 1/10], [31, 2/5], [32, 1/10], [33, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a3, [[30, 2/5], [31, 1/10], [32, 2/5], [33, 1/10]]], [b3, [[30, 1/10], [31, 2/5], [32, 1/10], [33, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0080","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8406c89ba505292520528a42d52f159e48eda4dabc19e072ab0e0b070b77089","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165080,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[40,41,42,43],\"statistic\":[0,1,0,1],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], No, [], [0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]], No, [], [0, [[a4, [[40, 3/4], [42, 1/4]]], [b4, [[40, 1/2], [42, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0081","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"110494fa232e3f2123ef775dc51cd119c52589afc8f66a86fdd807535c8f8095","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165081,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0082","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2955edd9adfcfd136378f26f08188acce4ba5ba5ced5ab9dfa8b853a23eb4730","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165082,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[60,61,62,63],\"statistic\":[0,1,2,3],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a6, b6], samples=[60, 61, 62, 63], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a6, [[60, 1]]], [b6, [[60, 1]]]], Yes], [1, [[a6, [[61, 1]]], [b6, [[61, 1]]]], Yes], [2, [[a6, [[62, 1]]], [b6, [[62, 1]]]], Yes], [3, [[a6, [[63, 1]]], [b6, [[63, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a6, [[60, 1]]], [b6, [[60, 1]]]], Yes], [1, [[a6, [[61, 1]]], [b6, [[61, 1]]]], Yes], [2, [[a6, [[62, 1]]], [b6, [[62, 1]]]], Yes], [3, [[a6, [[63, 1]]], [b6, [[63, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0083","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3547d24da3a8c7f3b08bb07aca67086296c4630055505420d89b0482fbd0120","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165083,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[70,71,72,73],\"statistic\":[0,0,0,0],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a7, b7], samples=[70, 71, 72, 73], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a7, [[70, 2/5], [71, 1/10], [72, 2/5], [73, 1/10]]], [b7, [[70, 1/10], [71, 2/5], [72, 1/10], [73, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a7, [[70, 2/5], [71, 1/10], [72, 2/5], [73, 1/10]]], [b7, [[70, 1/10], [71, 2/5], [72, 1/10], [73, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0084","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d4b4108cb35a77ce96549b284868c96ced8bffb1e6ca963939a9690a291298b","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165084,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[80,81,82,83],\"statistic\":[0,1,0,1],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a8, b8], samples=[80, 81, 82, 83], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], No, [], [0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], [1, [[a8, [[81, 1/2], [83, 1/2]]], [b8, [[81, 1/2], [83, 1/2]]]], Yes]], No, [], [0, [[a8, [[80, 3/4], [82, 1/4]]], [b8, [[80, 1/2], [82, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0085","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e4e9be4a59b61228292b93782955536100347e73df740da79a28a30e99bdbf","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165085,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[90,91,92,93],\"statistic\":[0,1,0,1],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a9, b9], samples=[90, 91, 92, 93], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, [[a9, 1, 0, null], [b9, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a9, [[90, 1/2], [92, 1/2]]]], Yes], [1, [[b9, [[91, 1/2], [93, 1/2]]]], Yes]], Yes, [[a9, 1, 0, null], [b9, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0086","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a11a164072e32ce7c8cb33fcf8eb243356528a56956fd420e86b88e2a216dfbd","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165086,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[100,101,102,103],\"statistic\":[0,1,2,3],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a10, b10], samples=[100, 101, 102, 103], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a10, [[100, 1]]], [b10, [[100, 1]]]], Yes], [1, [[a10, [[101, 1]]], [b10, [[101, 1]]]], Yes], [2, [[a10, [[102, 1]]], [b10, [[102, 1]]]], Yes], [3, [[a10, [[103, 1]]], [b10, [[103, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a10, [[100, 1]]], [b10, [[100, 1]]]], Yes], [1, [[a10, [[101, 1]]], [b10, [[101, 1]]]], Yes], [2, [[a10, [[102, 1]]], [b10, [[102, 1]]]], Yes], [3, [[a10, [[103, 1]]], [b10, [[103, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0087","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52306729e0fda83f63efae67b941b56474e903c68f2530a655246a7b3faf4431","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165087,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[110,111,112,113],\"statistic\":[0,0,0,0],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0088","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9037f6533ee6eb2941646039d56f79418fd090c795784cad74d01810f6d56b8b","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165088,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[120,121,122,123],\"statistic\":[0,1,0,1],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], No, [], [0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], No, [], [0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0089","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"189ad3bb990457caed20e55063b8e9dfe0c586b5bc115a698d5f9d2512e59958","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165089,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[130,131,132,133],\"statistic\":[0,1,0,1],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, [[a13, 1, 0, null], [b13, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, [[a13, 1, 0, null], [b13, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0090","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba241d1338c6bac0011871885139a30ecdfc6fc1838ea5c51a1dfeeb70df5fdc","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165090,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[140,141,142,143],\"statistic\":[0,1,2,3],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a14, b14], samples=[140, 141, 142, 143], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a14, [[140, 1]]], [b14, [[140, 1]]]], Yes], [1, [[a14, [[141, 1]]], [b14, [[141, 1]]]], Yes], [2, [[a14, [[142, 1]]], [b14, [[142, 1]]]], Yes], [3, [[a14, [[143, 1]]], [b14, [[143, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a14, [[140, 1]]], [b14, [[140, 1]]]], Yes], [1, [[a14, [[141, 1]]], [b14, [[141, 1]]]], Yes], [2, [[a14, [[142, 1]]], [b14, [[142, 1]]]], Yes], [3, [[a14, [[143, 1]]], [b14, [[143, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0091","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4a1b4a8ae9b0576a50eeb93298eb10431b9611a640e07196467ac0b6f737b9f","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165091,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[150,151,152,153],\"statistic\":[0,0,0,0],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a15, b15], samples=[150, 151, 152, 153], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a15, [[150, 2/5], [151, 1/10], [152, 2/5], [153, 1/10]]], [b15, [[150, 1/10], [151, 2/5], [152, 1/10], [153, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a15, [[150, 2/5], [151, 1/10], [152, 2/5], [153, 1/10]]], [b15, [[150, 1/10], [151, 2/5], [152, 1/10], [153, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0092","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8f04fe527142b59f2288fc98b4ee17d3b27d760ea6d9523fe326596e8bf373","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165092,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[160,161,162,163],\"statistic\":[0,1,0,1],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], No, [], [0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], [1, [[a16, [[161, 1/2], [163, 1/2]]], [b16, [[161, 1/2], [163, 1/2]]]], Yes]], No, [], [0, [[a16, [[160, 3/4], [162, 1/4]]], [b16, [[160, 1/2], [162, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0093","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f94ad5be4b04d8fc97b835d303e263e95cbe7e4f9f672e10f12bfe4605aea076","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165093,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, [[a17, 1, 0, null], [b17, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, [[a17, 1, 0, null], [b17, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0094","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d44491b7d3eb8530cbdb6ed69f035fc6262d91a2fd55257f905b1157d050934","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165094,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[180,181,182,183],\"statistic\":[0,1,2,3],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a18, b18], samples=[180, 181, 182, 183], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a18, [[180, 1]]], [b18, [[180, 1]]]], Yes], [1, [[a18, [[181, 1]]], [b18, [[181, 1]]]], Yes], [2, [[a18, [[182, 1]]], [b18, [[182, 1]]]], Yes], [3, [[a18, [[183, 1]]], [b18, [[183, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a18, [[180, 1]]], [b18, [[180, 1]]]], Yes], [1, [[a18, [[181, 1]]], [b18, [[181, 1]]]], Yes], [2, [[a18, [[182, 1]]], [b18, [[182, 1]]]], Yes], [3, [[a18, [[183, 1]]], [b18, [[183, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0095","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a3d45ba2abf88d82e15df8dd655b81872e745f13b4ff10add39c10dc44fb100","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165095,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[190,191,192,193],\"statistic\":[0,0,0,0],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a19, b19], samples=[190, 191, 192, 193], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a19, [[190, 2/5], [191, 1/10], [192, 2/5], [193, 1/10]]], [b19, [[190, 1/10], [191, 2/5], [192, 1/10], [193, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a19, [[190, 2/5], [191, 1/10], [192, 2/5], [193, 1/10]]], [b19, [[190, 1/10], [191, 2/5], [192, 1/10], [193, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0096","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b0f3aa245fe0c566d4feb8d5b8b842fd8567ee8a18870df52d7094bff7d3906","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165096,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[200,201,202,203],\"statistic\":[0,1,0,1],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], No, [], [0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], [1, [[a20, [[201, 1/2], [203, 1/2]]], [b20, [[201, 1/2], [203, 1/2]]]], Yes]], No, [], [0, [[a20, [[200, 3/4], [202, 1/4]]], [b20, [[200, 1/2], [202, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0097","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc989244b8cc07e4776bb52d0087323276d75047ac35da79c9b6d936160c403","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165097,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[210,211,212,213],\"statistic\":[0,1,0,1],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(zero_mass, [[0, [[a21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, [[a21, 1, 0, null], [b21, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, [[a21, 1, 0, null], [b21, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0098","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"890e53f12a755b0996f9a6fa9c7c1bbf9bce5f0282b7511882ef0eab700784a5","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165098,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"identity\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[220,221,222,223],\"statistic\":[0,1,2,3],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(identity, [[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (identity, [[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]], Yes, [], null, Yes, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0099","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18783ff4ea38911e8e2f058afb6ea7635cfebcfe7ed8940d58acdb05ac5d2ace","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165099,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"constant\",\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,0,0,0],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(constant, [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (constant, [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r05-finite-sufficient-statistic-0100","batch_id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3496eff962e0d2d111e0e2764f24bb7628596c7538d5ec477e591616f801be57","family_id":"qf-mod47-suff-nonsufficient-zero-boundary","seed":202611165100,"semantic_template":"qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"nonsufficient\",\"probability_rows\":[[[3,5],[1,10],[1,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[240,241,242,243],\"statistic\":[0,1,0,1],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks","stem":"Use complete theta labels=[a24, b24], samples=[240, 241, 242, 243], aligned exact probability rows=[[[3, 5], [1, 10], [1, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(nonsufficient, [[0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], No, [], [0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)","explanation":"Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (nonsufficient, [[0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], [1, [[a24, [[241, 1/2], [243, 1/2]]], [b24, [[241, 1/2], [243, 1/2]]]], Yes]], No, [], [0, [[a24, [[240, 3/4], [242, 1/4]]], [b24, [[240, 1/2], [242, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)."},"skills":["Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}