{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0001","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a0\",\"b0\"],\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x0\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, b0], [x0, y0], [[3/4, 1/4], [1/4, 3/4]], x0, [[a0, 3/4], [b0, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a0, b0], [x0, y0], [[3/4, 1/4], [1/4, 3/4]], x0, [[a0, 3/4], [b0, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164001,"parameters":{"theta_values":["a0","b0"],"observations":["x0","y0"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x0","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0002","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a1\",\"b1\"],\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x1\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, b1], [x1, y1], [[3/4, 1/4], [1/4, 3/4]], x1, [[a1, 3/4], [b1, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a1, b1], [x1, y1], [[3/4, 1/4], [1/4, 3/4]], x1, [[a1, 3/4], [b1, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164002,"parameters":{"theta_values":["a1","b1"],"observations":["x1","y1"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x1","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0003","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a2\",\"b2\"],\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x2\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, b2], [x2, y2], [[3/4, 1/4], [1/4, 3/4]], x2, [[a2, 3/4], [b2, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a2, b2], [x2, y2], [[3/4, 1/4], [1/4, 3/4]], x2, [[a2, 3/4], [b2, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164003,"parameters":{"theta_values":["a2","b2"],"observations":["x2","y2"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x2","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0004","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a3\",\"b3\"],\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x3\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, b3], [x3, y3], [[3/4, 1/4], [1/4, 3/4]], x3, [[a3, 3/4], [b3, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a3, b3], [x3, y3], [[3/4, 1/4], [1/4, 3/4]], x3, [[a3, 3/4], [b3, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164004,"parameters":{"theta_values":["a3","b3"],"observations":["x3","y3"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x3","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0005","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a4\",\"b4\"],\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x4\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, b4], [x4, y4], [[3/4, 1/4], [1/4, 3/4]], x4, [[a4, 3/4], [b4, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a4, b4], [x4, y4], [[3/4, 1/4], [1/4, 3/4]], x4, [[a4, 3/4], [b4, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164005,"parameters":{"theta_values":["a4","b4"],"observations":["x4","y4"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x4","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0006","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a5\",\"b5\"],\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x5\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, b5], [x5, y5], [[3/4, 1/4], [1/4, 3/4]], x5, [[a5, 3/4], [b5, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a5, b5], [x5, y5], [[3/4, 1/4], [1/4, 3/4]], x5, [[a5, 3/4], [b5, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164006,"parameters":{"theta_values":["a5","b5"],"observations":["x5","y5"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x5","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0007","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a6\",\"b6\"],\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x6\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, b6], [x6, y6], [[3/4, 1/4], [1/4, 3/4]], x6, [[a6, 3/4], [b6, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a6, b6], [x6, y6], [[3/4, 1/4], [1/4, 3/4]], x6, [[a6, 3/4], [b6, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164007,"parameters":{"theta_values":["a6","b6"],"observations":["x6","y6"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x6","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0008","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a7\",\"b7\"],\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x7\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, b7], [x7, y7], [[3/4, 1/4], [1/4, 3/4]], x7, [[a7, 3/4], [b7, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a7, b7], [x7, y7], [[3/4, 1/4], [1/4, 3/4]], x7, [[a7, 3/4], [b7, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164008,"parameters":{"theta_values":["a7","b7"],"observations":["x7","y7"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x7","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0009","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a8\",\"b8\"],\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x8\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, b8], [x8, y8], [[3/4, 1/4], [1/4, 3/4]], x8, [[a8, 3/4], [b8, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a8, b8], [x8, y8], [[3/4, 1/4], [1/4, 3/4]], x8, [[a8, 3/4], [b8, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164009,"parameters":{"theta_values":["a8","b8"],"observations":["x8","y8"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x8","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0010","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a9\",\"b9\"],\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x9\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, b9], [x9, y9], [[3/4, 1/4], [1/4, 3/4]], x9, [[a9, 3/4], [b9, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a9, b9], [x9, y9], [[3/4, 1/4], [1/4, 3/4]], x9, [[a9, 3/4], [b9, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164010,"parameters":{"theta_values":["a9","b9"],"observations":["x9","y9"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x9","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0011","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164011,"parameters":{"theta_values":["a10","b10"],"observations":["x10","y10"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x10","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0012","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a11\",\"b11\"],\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x11\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, b11], [x11, y11], [[3/4, 1/4], [1/4, 3/4]], x11, [[a11, 3/4], [b11, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a11, b11], [x11, y11], [[3/4, 1/4], [1/4, 3/4]], x11, [[a11, 3/4], [b11, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164012,"parameters":{"theta_values":["a11","b11"],"observations":["x11","y11"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x11","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0013","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a12\",\"b12\"],\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x12\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, b12], [x12, y12], [[3/4, 1/4], [1/4, 3/4]], x12, [[a12, 3/4], [b12, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a12, b12], [x12, y12], [[3/4, 1/4], [1/4, 3/4]], x12, [[a12, 3/4], [b12, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164013,"parameters":{"theta_values":["a12","b12"],"observations":["x12","y12"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x12","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0014","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a13\",\"b13\"],\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x13\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, b13], [x13, y13], [[3/4, 1/4], [1/4, 3/4]], x13, [[a13, 3/4], [b13, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a13, b13], [x13, y13], [[3/4, 1/4], [1/4, 3/4]], x13, [[a13, 3/4], [b13, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164014,"parameters":{"theta_values":["a13","b13"],"observations":["x13","y13"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x13","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0015","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a14\",\"b14\"],\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x14\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, b14], [x14, y14], [[3/4, 1/4], [1/4, 3/4]], x14, [[a14, 3/4], [b14, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a14, b14], [x14, y14], [[3/4, 1/4], [1/4, 3/4]], x14, [[a14, 3/4], [b14, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164015,"parameters":{"theta_values":["a14","b14"],"observations":["x14","y14"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x14","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0016","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a15\",\"b15\"],\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x15\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, b15], [x15, y15], [[3/4, 1/4], [1/4, 3/4]], x15, [[a15, 3/4], [b15, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a15, b15], [x15, y15], [[3/4, 1/4], [1/4, 3/4]], x15, [[a15, 3/4], [b15, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164016,"parameters":{"theta_values":["a15","b15"],"observations":["x15","y15"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x15","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0017","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, b16], [x16, y16], [[3/4, 1/4], [1/4, 3/4]], x16, [[a16, 3/4], [b16, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a16, b16], [x16, y16], [[3/4, 1/4], [1/4, 3/4]], x16, [[a16, 3/4], [b16, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164017,"parameters":{"theta_values":["a16","b16"],"observations":["x16","y16"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x16","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0018","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a17\",\"b17\"],\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x17\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, b17], [x17, y17], [[3/4, 1/4], [1/4, 3/4]], x17, [[a17, 3/4], [b17, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a17, b17], [x17, y17], [[3/4, 1/4], [1/4, 3/4]], x17, [[a17, 3/4], [b17, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164018,"parameters":{"theta_values":["a17","b17"],"observations":["x17","y17"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x17","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0019","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a18\",\"b18\"],\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x18\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, b18], [x18, y18], [[3/4, 1/4], [1/4, 3/4]], x18, [[a18, 3/4], [b18, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a18, b18], [x18, y18], [[3/4, 1/4], [1/4, 3/4]], x18, [[a18, 3/4], [b18, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164019,"parameters":{"theta_values":["a18","b18"],"observations":["x18","y18"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x18","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0020","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x19\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, b19], [x19, y19], [[3/4, 1/4], [1/4, 3/4]], x19, [[a19, 3/4], [b19, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a19, b19], [x19, y19], [[3/4, 1/4], [1/4, 3/4]], x19, [[a19, 3/4], [b19, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164020,"parameters":{"theta_values":["a19","b19"],"observations":["x19","y19"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x19","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0021","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a20\",\"b20\"],\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x20\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, b20], [x20, y20], [[3/4, 1/4], [1/4, 3/4]], x20, [[a20, 3/4], [b20, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a20, b20], [x20, y20], [[3/4, 1/4], [1/4, 3/4]], x20, [[a20, 3/4], [b20, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164021,"parameters":{"theta_values":["a20","b20"],"observations":["x20","y20"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x20","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0022","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a21\",\"b21\"],\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x21\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, b21], [x21, y21], [[3/4, 1/4], [1/4, 3/4]], x21, [[a21, 3/4], [b21, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a21, b21], [x21, y21], [[3/4, 1/4], [1/4, 3/4]], x21, [[a21, 3/4], [b21, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164022,"parameters":{"theta_values":["a21","b21"],"observations":["x21","y21"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x21","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0023","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a22\",\"b22\"],\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x22\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, b22], [x22, y22], [[3/4, 1/4], [1/4, 3/4]], x22, [[a22, 3/4], [b22, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a22, b22], [x22, y22], [[3/4, 1/4], [1/4, 3/4]], x22, [[a22, 3/4], [b22, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164023,"parameters":{"theta_values":["a22","b22"],"observations":["x22","y22"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x22","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0024","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a23\",\"b23\"],\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x23\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, b23], [x23, y23], [[3/4, 1/4], [1/4, 3/4]], x23, [[a23, 3/4], [b23, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a23, b23], [x23, y23], [[3/4, 1/4], [1/4, 3/4]], x23, [[a23, 3/4], [b23, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164024,"parameters":{"theta_values":["a23","b23"],"observations":["x23","y23"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x23","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0025","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-law-likelihood；資料={\"theta_values\":[\"a24\",\"b24\"],\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x24\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, b24], [x24, y24], [[3/4, 1/4], [1/4, 3/4]], x24, [[a24, 3/4], [b24, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation_zh":"Wave6第十層核對=([a24, b24], [x24, y24], [[3/4, 1/4], [1/4, 3/4]], x24, [[a24, 3/4], [b24, 1/4]], complete_normalized_finite_model_likelihood_table)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-law-likelihood","seed":202611164025,"parameters":{"theta_values":["a24","b24"],"observations":["x24","y24"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x24","domain":"Exact_finite_maximum_likelihood","operation":"mle-law-likelihood","semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0026","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a0\",\"b0\"],\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x0\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x0, [[a0, 3/4], [b0, 1/4]], 3/4, [a0], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x0, [[a0, 3/4], [b0, 1/4]], 3/4, [a0], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164026,"parameters":{"theta_values":["a0","b0"],"observations":["x0","y0"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x0","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0027","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a1\",\"b1\"],\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x1\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164027,"parameters":{"theta_values":["a1","b1"],"observations":["x1","y1"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x1","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0028","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a2\",\"b2\"],\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x2\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x2, [[a2, 3/4], [b2, 1/4]], 3/4, [a2], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x2, [[a2, 3/4], [b2, 1/4]], 3/4, [a2], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164028,"parameters":{"theta_values":["a2","b2"],"observations":["x2","y2"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x2","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0029","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a3\",\"b3\"],\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x3\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x3, [[a3, 3/4], [b3, 1/4]], 3/4, [a3], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x3, [[a3, 3/4], [b3, 1/4]], 3/4, [a3], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164029,"parameters":{"theta_values":["a3","b3"],"observations":["x3","y3"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x3","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0030","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a4\",\"b4\"],\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x4\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x4, [[a4, 3/4], [b4, 1/4]], 3/4, [a4], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x4, [[a4, 3/4], [b4, 1/4]], 3/4, [a4], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164030,"parameters":{"theta_values":["a4","b4"],"observations":["x4","y4"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x4","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0031","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a5\",\"b5\"],\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x5\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x5, [[a5, 3/4], [b5, 1/4]], 3/4, [a5], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x5, [[a5, 3/4], [b5, 1/4]], 3/4, [a5], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164031,"parameters":{"theta_values":["a5","b5"],"observations":["x5","y5"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x5","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0032","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a6\",\"b6\"],\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x6\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x6, [[a6, 3/4], [b6, 1/4]], 3/4, [a6], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x6, [[a6, 3/4], [b6, 1/4]], 3/4, [a6], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164032,"parameters":{"theta_values":["a6","b6"],"observations":["x6","y6"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x6","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0033","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a7\",\"b7\"],\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x7\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x7, [[a7, 3/4], [b7, 1/4]], 3/4, [a7], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x7, [[a7, 3/4], [b7, 1/4]], 3/4, [a7], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164033,"parameters":{"theta_values":["a7","b7"],"observations":["x7","y7"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x7","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0034","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a8\",\"b8\"],\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x8\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x8, [[a8, 3/4], [b8, 1/4]], 3/4, [a8], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x8, [[a8, 3/4], [b8, 1/4]], 3/4, [a8], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164034,"parameters":{"theta_values":["a8","b8"],"observations":["x8","y8"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x8","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0035","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a9\",\"b9\"],\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x9\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x9, [[a9, 3/4], [b9, 1/4]], 3/4, [a9], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x9, [[a9, 3/4], [b9, 1/4]], 3/4, [a9], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164035,"parameters":{"theta_values":["a9","b9"],"observations":["x9","y9"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x9","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0036","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x10, [[a10, 3/4], [b10, 1/4]], 3/4, [a10], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x10, [[a10, 3/4], [b10, 1/4]], 3/4, [a10], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164036,"parameters":{"theta_values":["a10","b10"],"observations":["x10","y10"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x10","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0037","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a11\",\"b11\"],\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x11\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x11, [[a11, 3/4], [b11, 1/4]], 3/4, [a11], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x11, [[a11, 3/4], [b11, 1/4]], 3/4, [a11], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164037,"parameters":{"theta_values":["a11","b11"],"observations":["x11","y11"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x11","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0038","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a12\",\"b12\"],\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x12\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164038,"parameters":{"theta_values":["a12","b12"],"observations":["x12","y12"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x12","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0039","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a13\",\"b13\"],\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x13\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x13, [[a13, 3/4], [b13, 1/4]], 3/4, [a13], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x13, [[a13, 3/4], [b13, 1/4]], 3/4, [a13], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164039,"parameters":{"theta_values":["a13","b13"],"observations":["x13","y13"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x13","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0040","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a14\",\"b14\"],\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x14\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x14, [[a14, 3/4], [b14, 1/4]], 3/4, [a14], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x14, [[a14, 3/4], [b14, 1/4]], 3/4, [a14], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164040,"parameters":{"theta_values":["a14","b14"],"observations":["x14","y14"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x14","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0041","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a15\",\"b15\"],\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x15\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x15, [[a15, 3/4], [b15, 1/4]], 3/4, [a15], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x15, [[a15, 3/4], [b15, 1/4]], 3/4, [a15], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164041,"parameters":{"theta_values":["a15","b15"],"observations":["x15","y15"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x15","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164042,"parameters":{"theta_values":["a16","b16"],"observations":["x16","y16"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x16","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0043","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a17\",\"b17\"],\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x17\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x17, [[a17, 3/4], [b17, 1/4]], 3/4, [a17], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x17, [[a17, 3/4], [b17, 1/4]], 3/4, [a17], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164043,"parameters":{"theta_values":["a17","b17"],"observations":["x17","y17"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x17","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0044","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a18\",\"b18\"],\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x18\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x18, [[a18, 3/4], [b18, 1/4]], 3/4, [a18], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x18, [[a18, 3/4], [b18, 1/4]], 3/4, [a18], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164044,"parameters":{"theta_values":["a18","b18"],"observations":["x18","y18"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x18","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0045","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x19\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x19, [[a19, 3/4], [b19, 1/4]], 3/4, [a19], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x19, [[a19, 3/4], [b19, 1/4]], 3/4, [a19], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164045,"parameters":{"theta_values":["a19","b19"],"observations":["x19","y19"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x19","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0046","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a20\",\"b20\"],\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x20\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x20, [[a20, 3/4], [b20, 1/4]], 3/4, [a20], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x20, [[a20, 3/4], [b20, 1/4]], 3/4, [a20], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164046,"parameters":{"theta_values":["a20","b20"],"observations":["x20","y20"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x20","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0047","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a21\",\"b21\"],\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x21\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x21, [[a21, 3/4], [b21, 1/4]], 3/4, [a21], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x21, [[a21, 3/4], [b21, 1/4]], 3/4, [a21], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164047,"parameters":{"theta_values":["a21","b21"],"observations":["x21","y21"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x21","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0048","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a22\",\"b22\"],\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x22\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x22, [[a22, 3/4], [b22, 1/4]], 3/4, [a22], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x22, [[a22, 3/4], [b22, 1/4]], 3/4, [a22], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164048,"parameters":{"theta_values":["a22","b22"],"observations":["x22","y22"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x22","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0049","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a23\",\"b23\"],\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x23\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x23, [[a23, 3/4], [b23, 1/4]], 3/4, [a23], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x23, [[a23, 3/4], [b23, 1/4]], 3/4, [a23], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164049,"parameters":{"theta_values":["a23","b23"],"observations":["x23","y23"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x23","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0050","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-argmax；資料={\"theta_values\":[\"a24\",\"b24\"],\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x24\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x24, [[a24, 3/4], [b24, 1/4]], 3/4, [a24], 是, MLE_argmax_need_not_be_unique)","explanation_zh":"Wave6第十層核對=(x24, [[a24, 3/4], [b24, 1/4]], 3/4, [a24], 是, MLE_argmax_need_not_be_unique)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-argmax","seed":202611164050,"parameters":{"theta_values":["a24","b24"],"observations":["x24","y24"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"observed":"x24","domain":"Exact_finite_maximum_likelihood","operation":"mle-argmax","semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0051","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a0\",\"b0\"],\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a0\",\"b0\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164051,"parameters":{"theta_values":["a0","b0"],"observations":["x0","y0"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a0","b0"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"],\"tie_order\":[\"a0\",\"b0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0052","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a1\",\"b1\"],\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a1\",\"b1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, b1], [[x1, [a1], a1], [y1, [b1], b1]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a1, b1], [[x1, [a1], a1], [y1, [b1], b1]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164052,"parameters":{"theta_values":["a1","b1"],"observations":["x1","y1"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a1","b1"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"],\"tie_order\":[\"a1\",\"b1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0053","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a2\",\"b2\"],\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a2\",\"b2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, b2], [[x2, [a2], a2], [y2, [b2], b2]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a2, b2], [[x2, [a2], a2], [y2, [b2], b2]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164053,"parameters":{"theta_values":["a2","b2"],"observations":["x2","y2"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a2","b2"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"],\"tie_order\":[\"a2\",\"b2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0054","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a3\",\"b3\"],\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a3\",\"b3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, b3], [[x3, [a3], a3], [y3, [b3], b3]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a3, b3], [[x3, [a3], a3], [y3, [b3], b3]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164054,"parameters":{"theta_values":["a3","b3"],"observations":["x3","y3"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a3","b3"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"],\"tie_order\":[\"a3\",\"b3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0055","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a4\",\"b4\"],\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a4\",\"b4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, b4], [[x4, [a4], a4], [y4, [b4], b4]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a4, b4], [[x4, [a4], a4], [y4, [b4], b4]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164055,"parameters":{"theta_values":["a4","b4"],"observations":["x4","y4"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a4","b4"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"],\"tie_order\":[\"a4\",\"b4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0056","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a5\",\"b5\"],\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a5\",\"b5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, b5], [[x5, [a5], a5], [y5, [b5], b5]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a5, b5], [[x5, [a5], a5], [y5, [b5], b5]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164056,"parameters":{"theta_values":["a5","b5"],"observations":["x5","y5"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a5","b5"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"],\"tie_order\":[\"a5\",\"b5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0057","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a6\",\"b6\"],\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a6\",\"b6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, b6], [[x6, [a6], a6], [y6, [b6], b6]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a6, b6], [[x6, [a6], a6], [y6, [b6], b6]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164057,"parameters":{"theta_values":["a6","b6"],"observations":["x6","y6"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a6","b6"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"],\"tie_order\":[\"a6\",\"b6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0058","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a7\",\"b7\"],\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a7\",\"b7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, b7], [[x7, [a7], a7], [y7, [b7], b7]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a7, b7], [[x7, [a7], a7], [y7, [b7], b7]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164058,"parameters":{"theta_values":["a7","b7"],"observations":["x7","y7"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a7","b7"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"],\"tie_order\":[\"a7\",\"b7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0059","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a8\",\"b8\"],\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a8\",\"b8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, b8], [[x8, [a8], a8], [y8, [b8], b8]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a8, b8], [[x8, [a8], a8], [y8, [b8], b8]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164059,"parameters":{"theta_values":["a8","b8"],"observations":["x8","y8"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a8","b8"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"],\"tie_order\":[\"a8\",\"b8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0060","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a9\",\"b9\"],\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a9\",\"b9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, b9], [[x9, [a9], a9], [y9, [b9], b9]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a9, b9], [[x9, [a9], a9], [y9, [b9], b9]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164060,"parameters":{"theta_values":["a9","b9"],"observations":["x9","y9"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a9","b9"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"],\"tie_order\":[\"a9\",\"b9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0061","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a10\",\"b10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164061,"parameters":{"theta_values":["a10","b10"],"observations":["x10","y10"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a10","b10"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"],\"tie_order\":[\"a10\",\"b10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0062","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a11\",\"b11\"],\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a11\",\"b11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, b11], [[x11, [a11], a11], [y11, [b11], b11]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a11, b11], [[x11, [a11], a11], [y11, [b11], b11]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164062,"parameters":{"theta_values":["a11","b11"],"observations":["x11","y11"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a11","b11"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"],\"tie_order\":[\"a11\",\"b11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0063","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a12\",\"b12\"],\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a12\",\"b12\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, b12], [[x12, [a12], a12], [y12, [b12], b12]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a12, b12], [[x12, [a12], a12], [y12, [b12], b12]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164063,"parameters":{"theta_values":["a12","b12"],"observations":["x12","y12"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a12","b12"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"],\"tie_order\":[\"a12\",\"b12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0064","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a13\",\"b13\"],\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a13\",\"b13\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164064,"parameters":{"theta_values":["a13","b13"],"observations":["x13","y13"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a13","b13"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"],\"tie_order\":[\"a13\",\"b13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0065","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a14\",\"b14\"],\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a14\",\"b14\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, b14], [[x14, [a14], a14], [y14, [b14], b14]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a14, b14], [[x14, [a14], a14], [y14, [b14], b14]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164065,"parameters":{"theta_values":["a14","b14"],"observations":["x14","y14"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a14","b14"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"],\"tie_order\":[\"a14\",\"b14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0066","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a15\",\"b15\"],\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a15\",\"b15\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, b15], [[x15, [a15], a15], [y15, [b15], b15]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a15, b15], [[x15, [a15], a15], [y15, [b15], b15]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164066,"parameters":{"theta_values":["a15","b15"],"observations":["x15","y15"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a15","b15"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"],\"tie_order\":[\"a15\",\"b15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0067","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a16\",\"b16\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, b16], [[x16, [a16], a16], [y16, [b16], b16]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a16, b16], [[x16, [a16], a16], [y16, [b16], b16]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164067,"parameters":{"theta_values":["a16","b16"],"observations":["x16","y16"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a16","b16"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"],\"tie_order\":[\"a16\",\"b16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0068","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a17\",\"b17\"],\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a17\",\"b17\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, b17], [[x17, [a17], a17], [y17, [b17], b17]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a17, b17], [[x17, [a17], a17], [y17, [b17], b17]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164068,"parameters":{"theta_values":["a17","b17"],"observations":["x17","y17"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a17","b17"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"],\"tie_order\":[\"a17\",\"b17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0069","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a18\",\"b18\"],\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a18\",\"b18\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164069,"parameters":{"theta_values":["a18","b18"],"observations":["x18","y18"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a18","b18"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"],\"tie_order\":[\"a18\",\"b18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0070","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, b19], [[x19, [a19], a19], [y19, [b19], b19]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a19, b19], [[x19, [a19], a19], [y19, [b19], b19]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164070,"parameters":{"theta_values":["a19","b19"],"observations":["x19","y19"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a19","b19"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0071","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a20\",\"b20\"],\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a20\",\"b20\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, b20], [[x20, [a20], a20], [y20, [b20], b20]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a20, b20], [[x20, [a20], a20], [y20, [b20], b20]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164071,"parameters":{"theta_values":["a20","b20"],"observations":["x20","y20"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a20","b20"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"],\"tie_order\":[\"a20\",\"b20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0072","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a21\",\"b21\"],\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a21\",\"b21\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164072,"parameters":{"theta_values":["a21","b21"],"observations":["x21","y21"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a21","b21"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"],\"tie_order\":[\"a21\",\"b21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0073","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a22\",\"b22\"],\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a22\",\"b22\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, b22], [[x22, [a22], a22], [y22, [b22], b22]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a22, b22], [[x22, [a22], a22], [y22, [b22], b22]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164073,"parameters":{"theta_values":["a22","b22"],"observations":["x22","y22"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a22","b22"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"],\"tie_order\":[\"a22\",\"b22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0074","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a23\",\"b23\"],\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a23\",\"b23\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, b23], [[x23, [a23], a23], [y23, [b23], b23]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a23, b23], [[x23, [a23], a23], [y23, [b23], b23]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164074,"parameters":{"theta_values":["a23","b23"],"observations":["x23","y23"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a23","b23"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"],\"tie_order\":[\"a23\",\"b23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0075","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-total-tie-estimator；資料={\"theta_values\":[\"a24\",\"b24\"],\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"tie_order\":[\"a24\",\"b24\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, b24], [[x24, [a24], a24], [y24, [b24], b24]], 是, theta_independent_tie_rule_makes_a_total_estimator)","explanation_zh":"Wave6第十層核對=([a24, b24], [[x24, [a24], a24], [y24, [b24], b24]], 是, theta_independent_tie_rule_makes_a_total_estimator)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164075,"parameters":{"theta_values":["a24","b24"],"observations":["x24","y24"],"probability_rows":[[[3,4],[1,4]],[[1,4],[3,4]]],"tie_order":["a24","b24"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-total-tie-estimator","semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"],\"tie_order\":[\"a24\",\"b24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0076","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a0\",\"b0\"],\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x0\",\"tie_order\":[\"a0\",\"b0\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x0, [[a0, 0], [b0, 0]], 0, [a0, b0], [a0, b0], [x0, [a0, b0], a0], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x0, [[a0, 0], [b0, 0]], 0, [a0, b0], [a0, b0], [x0, [a0, b0], a0], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164076,"parameters":{"mode":"all_zero_likelihood","theta_values":["a0","b0"],"observations":["x0","y0"],"probability_rows":[[0,1],[0,1]],"observed":"x0","tie_order":["a0","b0"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a0\",\"b0\"],\"tie_order\":[\"a0\",\"b0\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0077","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a1\",\"b1\"],\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x1\",\"tie_order\":[\"a1\",\"b1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x1, [[a1, 1/2], [b1, 1/2]], 1/2, [a1, b1], [a1, b1], [x1, [a1, b1], a1], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x1, [[a1, 1/2], [b1, 1/2]], 1/2, [a1, b1], [a1, b1], [x1, [a1, b1], a1], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164077,"parameters":{"mode":"tie","theta_values":["a1","b1"],"observations":["x1","y1"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x1","tie_order":["a1","b1"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a1\",\"b1\"],\"tie_order\":[\"a1\",\"b1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0078","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a2\",\"b2\"],\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x2\",\"tie_order\":[\"a2\",\"b2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x2, [[a2, 0], [b2, 0]], 0, [a2, b2], [a2, b2], [x2, [a2, b2], a2], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x2, [[a2, 0], [b2, 0]], 0, [a2, b2], [a2, b2], [x2, [a2, b2], a2], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164078,"parameters":{"mode":"all_zero_likelihood","theta_values":["a2","b2"],"observations":["x2","y2"],"probability_rows":[[0,1],[0,1]],"observed":"x2","tie_order":["a2","b2"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a2\",\"b2\"],\"tie_order\":[\"a2\",\"b2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0079","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a3\",\"b3\"],\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x3\",\"tie_order\":[\"a3\",\"b3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x3, [[a3, 1/2], [b3, 1/2]], 1/2, [a3, b3], [a3, b3], [x3, [a3, b3], a3], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x3, [[a3, 1/2], [b3, 1/2]], 1/2, [a3, b3], [a3, b3], [x3, [a3, b3], a3], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164079,"parameters":{"mode":"tie","theta_values":["a3","b3"],"observations":["x3","y3"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x3","tie_order":["a3","b3"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a3\",\"b3\"],\"tie_order\":[\"a3\",\"b3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0080","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a4\",\"b4\"],\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x4\",\"tie_order\":[\"a4\",\"b4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x4, [[a4, 0], [b4, 0]], 0, [a4, b4], [a4, b4], [x4, [a4, b4], a4], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x4, [[a4, 0], [b4, 0]], 0, [a4, b4], [a4, b4], [x4, [a4, b4], a4], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164080,"parameters":{"mode":"all_zero_likelihood","theta_values":["a4","b4"],"observations":["x4","y4"],"probability_rows":[[0,1],[0,1]],"observed":"x4","tie_order":["a4","b4"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a4\",\"b4\"],\"tie_order\":[\"a4\",\"b4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0081","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a5\",\"b5\"],\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x5\",\"tie_order\":[\"a5\",\"b5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x5, [[a5, 1/2], [b5, 1/2]], 1/2, [a5, b5], [a5, b5], [x5, [a5, b5], a5], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x5, [[a5, 1/2], [b5, 1/2]], 1/2, [a5, b5], [a5, b5], [x5, [a5, b5], a5], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164081,"parameters":{"mode":"tie","theta_values":["a5","b5"],"observations":["x5","y5"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x5","tie_order":["a5","b5"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a5\",\"b5\"],\"tie_order\":[\"a5\",\"b5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0082","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a6\",\"b6\"],\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x6\",\"tie_order\":[\"a6\",\"b6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x6, [[a6, 0], [b6, 0]], 0, [a6, b6], [a6, b6], [x6, [a6, b6], a6], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x6, [[a6, 0], [b6, 0]], 0, [a6, b6], [a6, b6], [x6, [a6, b6], a6], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164082,"parameters":{"mode":"all_zero_likelihood","theta_values":["a6","b6"],"observations":["x6","y6"],"probability_rows":[[0,1],[0,1]],"observed":"x6","tie_order":["a6","b6"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a6\",\"b6\"],\"tie_order\":[\"a6\",\"b6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0083","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a7\",\"b7\"],\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x7\",\"tie_order\":[\"a7\",\"b7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x7, [[a7, 1/2], [b7, 1/2]], 1/2, [a7, b7], [a7, b7], [x7, [a7, b7], a7], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x7, [[a7, 1/2], [b7, 1/2]], 1/2, [a7, b7], [a7, b7], [x7, [a7, b7], a7], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164083,"parameters":{"mode":"tie","theta_values":["a7","b7"],"observations":["x7","y7"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x7","tie_order":["a7","b7"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a7\",\"b7\"],\"tie_order\":[\"a7\",\"b7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0084","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a8\",\"b8\"],\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x8\",\"tie_order\":[\"a8\",\"b8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x8, [[a8, 0], [b8, 0]], 0, [a8, b8], [a8, b8], [x8, [a8, b8], a8], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x8, [[a8, 0], [b8, 0]], 0, [a8, b8], [a8, b8], [x8, [a8, b8], a8], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164084,"parameters":{"mode":"all_zero_likelihood","theta_values":["a8","b8"],"observations":["x8","y8"],"probability_rows":[[0,1],[0,1]],"observed":"x8","tie_order":["a8","b8"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a8\",\"b8\"],\"tie_order\":[\"a8\",\"b8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0085","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a9\",\"b9\"],\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x9\",\"tie_order\":[\"a9\",\"b9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x9, [[a9, 1/2], [b9, 1/2]], 1/2, [a9, b9], [a9, b9], [x9, [a9, b9], a9], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x9, [[a9, 1/2], [b9, 1/2]], 1/2, [a9, b9], [a9, b9], [x9, [a9, b9], a9], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164085,"parameters":{"mode":"tie","theta_values":["a9","b9"],"observations":["x9","y9"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x9","tie_order":["a9","b9"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a9\",\"b9\"],\"tie_order\":[\"a9\",\"b9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0086","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x10\",\"tie_order\":[\"a10\",\"b10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x10, [[a10, 0], [b10, 0]], 0, [a10, b10], [a10, b10], [x10, [a10, b10], a10], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x10, [[a10, 0], [b10, 0]], 0, [a10, b10], [a10, b10], [x10, [a10, b10], a10], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164086,"parameters":{"mode":"all_zero_likelihood","theta_values":["a10","b10"],"observations":["x10","y10"],"probability_rows":[[0,1],[0,1]],"observed":"x10","tie_order":["a10","b10"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a10\",\"b10\"],\"tie_order\":[\"a10\",\"b10\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0087","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a11\",\"b11\"],\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x11\",\"tie_order\":[\"a11\",\"b11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x11, [[a11, 1/2], [b11, 1/2]], 1/2, [a11, b11], [a11, b11], [x11, [a11, b11], a11], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x11, [[a11, 1/2], [b11, 1/2]], 1/2, [a11, b11], [a11, b11], [x11, [a11, b11], a11], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164087,"parameters":{"mode":"tie","theta_values":["a11","b11"],"observations":["x11","y11"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x11","tie_order":["a11","b11"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a11\",\"b11\"],\"tie_order\":[\"a11\",\"b11\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0088","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a12\",\"b12\"],\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x12\",\"tie_order\":[\"a12\",\"b12\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x12, [[a12, 0], [b12, 0]], 0, [a12, b12], [a12, b12], [x12, [a12, b12], a12], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x12, [[a12, 0], [b12, 0]], 0, [a12, b12], [a12, b12], [x12, [a12, b12], a12], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164088,"parameters":{"mode":"all_zero_likelihood","theta_values":["a12","b12"],"observations":["x12","y12"],"probability_rows":[[0,1],[0,1]],"observed":"x12","tie_order":["a12","b12"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a12\",\"b12\"],\"tie_order\":[\"a12\",\"b12\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0089","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a13\",\"b13\"],\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x13\",\"tie_order\":[\"a13\",\"b13\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x13, [[a13, 1/2], [b13, 1/2]], 1/2, [a13, b13], [a13, b13], [x13, [a13, b13], a13], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x13, [[a13, 1/2], [b13, 1/2]], 1/2, [a13, b13], [a13, b13], [x13, [a13, b13], a13], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164089,"parameters":{"mode":"tie","theta_values":["a13","b13"],"observations":["x13","y13"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x13","tie_order":["a13","b13"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a13\",\"b13\"],\"tie_order\":[\"a13\",\"b13\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0090","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a14\",\"b14\"],\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x14\",\"tie_order\":[\"a14\",\"b14\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x14, [[a14, 0], [b14, 0]], 0, [a14, b14], [a14, b14], [x14, [a14, b14], a14], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x14, [[a14, 0], [b14, 0]], 0, [a14, b14], [a14, b14], [x14, [a14, b14], a14], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164090,"parameters":{"mode":"all_zero_likelihood","theta_values":["a14","b14"],"observations":["x14","y14"],"probability_rows":[[0,1],[0,1]],"observed":"x14","tie_order":["a14","b14"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a14\",\"b14\"],\"tie_order\":[\"a14\",\"b14\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0091","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a15\",\"b15\"],\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x15\",\"tie_order\":[\"a15\",\"b15\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x15, [[a15, 1/2], [b15, 1/2]], 1/2, [a15, b15], [a15, b15], [x15, [a15, b15], a15], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x15, [[a15, 1/2], [b15, 1/2]], 1/2, [a15, b15], [a15, b15], [x15, [a15, b15], a15], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164091,"parameters":{"mode":"tie","theta_values":["a15","b15"],"observations":["x15","y15"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x15","tie_order":["a15","b15"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a15\",\"b15\"],\"tie_order\":[\"a15\",\"b15\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0092","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x16\",\"tie_order\":[\"a16\",\"b16\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x16, [[a16, 0], [b16, 0]], 0, [a16, b16], [a16, b16], [x16, [a16, b16], a16], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x16, [[a16, 0], [b16, 0]], 0, [a16, b16], [a16, b16], [x16, [a16, b16], a16], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164092,"parameters":{"mode":"all_zero_likelihood","theta_values":["a16","b16"],"observations":["x16","y16"],"probability_rows":[[0,1],[0,1]],"observed":"x16","tie_order":["a16","b16"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a16\",\"b16\"],\"tie_order\":[\"a16\",\"b16\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0093","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a17\",\"b17\"],\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x17\",\"tie_order\":[\"a17\",\"b17\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x17, [[a17, 1/2], [b17, 1/2]], 1/2, [a17, b17], [a17, b17], [x17, [a17, b17], a17], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x17, [[a17, 1/2], [b17, 1/2]], 1/2, [a17, b17], [a17, b17], [x17, [a17, b17], a17], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164093,"parameters":{"mode":"tie","theta_values":["a17","b17"],"observations":["x17","y17"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x17","tie_order":["a17","b17"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a17\",\"b17\"],\"tie_order\":[\"a17\",\"b17\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0094","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a18\",\"b18\"],\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x18\",\"tie_order\":[\"a18\",\"b18\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x18, [[a18, 0], [b18, 0]], 0, [a18, b18], [a18, b18], [x18, [a18, b18], a18], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x18, [[a18, 0], [b18, 0]], 0, [a18, b18], [a18, b18], [x18, [a18, b18], a18], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164094,"parameters":{"mode":"all_zero_likelihood","theta_values":["a18","b18"],"observations":["x18","y18"],"probability_rows":[[0,1],[0,1]],"observed":"x18","tie_order":["a18","b18"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a18\",\"b18\"],\"tie_order\":[\"a18\",\"b18\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0095","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x19\",\"tie_order\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164095,"parameters":{"mode":"tie","theta_values":["a19","b19"],"observations":["x19","y19"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x19","tie_order":["a19","b19"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0096","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a20\",\"b20\"],\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x20\",\"tie_order\":[\"a20\",\"b20\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x20, [[a20, 0], [b20, 0]], 0, [a20, b20], [a20, b20], [x20, [a20, b20], a20], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x20, [[a20, 0], [b20, 0]], 0, [a20, b20], [a20, b20], [x20, [a20, b20], a20], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164096,"parameters":{"mode":"all_zero_likelihood","theta_values":["a20","b20"],"observations":["x20","y20"],"probability_rows":[[0,1],[0,1]],"observed":"x20","tie_order":["a20","b20"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a20\",\"b20\"],\"tie_order\":[\"a20\",\"b20\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0097","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a21\",\"b21\"],\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x21\",\"tie_order\":[\"a21\",\"b21\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x21, [[a21, 1/2], [b21, 1/2]], 1/2, [a21, b21], [a21, b21], [x21, [a21, b21], a21], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x21, [[a21, 1/2], [b21, 1/2]], 1/2, [a21, b21], [a21, b21], [x21, [a21, b21], a21], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164097,"parameters":{"mode":"tie","theta_values":["a21","b21"],"observations":["x21","y21"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x21","tie_order":["a21","b21"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a21\",\"b21\"],\"tie_order\":[\"a21\",\"b21\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0098","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a22\",\"b22\"],\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x22\",\"tie_order\":[\"a22\",\"b22\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x22, [[a22, 0], [b22, 0]], 0, [a22, b22], [a22, b22], [x22, [a22, b22], a22], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x22, [[a22, 0], [b22, 0]], 0, [a22, b22], [a22, b22], [x22, [a22, b22], a22], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164098,"parameters":{"mode":"all_zero_likelihood","theta_values":["a22","b22"],"observations":["x22","y22"],"probability_rows":[[0,1],[0,1]],"observed":"x22","tie_order":["a22","b22"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a22\",\"b22\"],\"tie_order\":[\"a22\",\"b22\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0099","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a23\",\"b23\"],\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x23\",\"tie_order\":[\"a23\",\"b23\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(tie, x23, [[a23, 1/2], [b23, 1/2]], 1/2, [a23, b23], [a23, b23], [x23, [a23, b23], a23], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(tie, x23, [[a23, 1/2], [b23, 1/2]], 1/2, [a23, b23], [a23, b23], [x23, [a23, b23], a23], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164099,"parameters":{"mode":"tie","theta_values":["a23","b23"],"observations":["x23","y23"],"probability_rows":[[[1,2],[1,2]],[[1,2],[1,2]]],"observed":"x23","tie_order":["a23","b23"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a23\",\"b23\"],\"tie_order\":[\"a23\",\"b23\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0100","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-maximum-likelihood-estimator"],"title_zh":"有限模型最大似然估計量","stem_zh":"完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"all_zero_likelihood\",\"theta_values\":[\"a24\",\"b24\"],\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[0,1],[0,1]],\"observed\":\"x24\",\"tie_order\":[\"a24\",\"b24\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(all_zero_likelihood, x24, [[a24, 0], [b24, 0]], 0, [a24, b24], [a24, b24], [x24, [a24, b24], a24], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation_zh":"Wave6第十層核對=(all_zero_likelihood, x24, [[a24, 0], [b24, 0]], 0, [a24, b24], [a24, b24], [x24, [a24, b24], a24], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限模型最大似然估計量"]},"variant":{"family_id":"qf-mod47-mle-boundary","seed":202611164100,"parameters":{"mode":"all_zero_likelihood","theta_values":["a24","b24"],"observations":["x24","y24"],"probability_rows":[[0,1],[0,1]],"observed":"x24","tie_order":["a24","b24"],"domain":"Exact_finite_maximum_likelihood","operation":"mle-boundary","semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a24\",\"b24\"],\"tie_order\":[\"a24\",\"b24\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}