{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0024",
  "pilot_number": "P1000-0649",
  "number": 649,
  "family_id": "qf-mod47-suff-fibers",
  "title": {
    "zh-Hant": "第 649 題：有限模型充分統計量",
    "en": "Question 649: A Total Statistic Partitions the Complete Sample Carrier into Fibers"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0024",
    "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-sufficient-statistic"
    ],
    "title_zh": "有限模型充分統計量",
    "stem_zh": "total statistic fibers與正質量conditional laws的完整theta-invariance責任。操作=suff-fibers；資料={\"theta_values\":[\"a23\",\"b23\"],\"samples\":[230,231,232,233],\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], 是, total_statistic_fibers_partition_sample_space)",
      "explanation_zh": "Wave6第十層核對=([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], 是, total_statistic_fibers_partition_sample_space)；依完整明示資料獨立重算；zero-mass conditional保持undefined，nonsufficient需明示witness；identity/constant boundaries分開且不使用factorization theorem或宣稱估計品質。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限模型充分統計量"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-suff-fibers",
      "seed": 202611165024,
      "parameters": {
        "theta_values": [
          "a23",
          "b23"
        ],
        "samples": [
          230,
          231,
          232,
          233
        ],
        "probability_rows": [
          [
            [
              2,
              5
            ],
            [
              1,
              10
            ],
            [
              2,
              5
            ],
            [
              1,
              10
            ]
          ],
          [
            [
              1,
              10
            ],
            [
              2,
              5
            ],
            [
              1,
              10
            ],
            [
              2,
              5
            ]
          ]
        ],
        "statistic": [
          0,
          1,
          0,
          1
        ],
        "domain": "Exact_finite_sufficient_statistic",
        "operation": "suff-fibers",
        "semantic_template": "qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,1,0,1],\"theta_values\":[\"a23\",\"b23\"]}"
      }
    },
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0024",
    "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d7a1a14f75465ea1b170991136027d953a21afe3951b4b762d72b7d73b29b907",
      "family_id": "qf-mod47-suff-fibers",
      "seed": 202611165024,
      "semantic_template": "qf-mod47-suff-fibers/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[230,231,232,233],\"statistic\":[0,1,0,1],\"theta_values\":[\"a23\",\"b23\"]}"
    },
    "fields": {
      "title": "A Total Statistic Partitions the Complete Sample Carrier into Fibers",
      "stem": "Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space)",
        "explanation": "Sort samples with their statistic values and laws. List every statistic fiber and compare its total covered count with the sample count. A partition alone does not establish theta-independent conditional laws or estimator quality. The exact result, retaining the requested order, is ([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], Yes, total_statistic_fibers_partition_sample_space)."
      },
      "skills": [
        "A Total Statistic Partitions the Complete Sample Carrier into Fibers"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "total statistic fibers與正質量conditional laws的完整theta-invariance責任。操作=suff-fibers；資料={\"theta_values\":[\"a23\",\"b23\"],\"samples\":[230,231,232,233],\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Θ=[a23, b23]; Ω=[230, 231, 232, 233]",
      "after": "[[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]]",
      "relation": "implies",
      "rule_id": "full_parametric_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列是一個固定參數的law，每欄是一個樣本結果；重排標籤時同時重排機率。",
        "en": "Each row is one parameter law and each column an outcome; reorder probabilities together with labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "P_a23(Ω) := sum(2/5, 1/10, 2/5, 1/10)",
      "after": "P_a23(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對此law的完整總質量。",
        "en": "Check this law's complete total mass first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_a23(Ω)，不以小數近似替代。",
          "en": "Recompute P_a23(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2/5",
          "1/10",
          "2/5",
          "1/10"
        ],
        "output": "1",
        "label": "P_a23(Ω)"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "P_b23(Ω) := sum(1/10, 2/5, 1/10, 2/5)",
      "after": "P_b23(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對此law的完整總質量。",
        "en": "Check this law's complete total mass first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_b23(Ω)，不以小數近似替代。",
          "en": "Recompute P_b23(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/10",
          "2/5",
          "1/10",
          "2/5"
        ],
        "output": "1",
        "label": "P_b23(Ω)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "sample=230",
      "after": "T(230)=0",
      "relation": "implies",
      "rule_id": "total_statistic_map",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個樣本點都恰有一個統計量值。",
        "en": "Every sample has exactly one statistic value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "sample=231",
      "after": "T(231)=1",
      "relation": "implies",
      "rule_id": "total_statistic_map",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個樣本點都恰有一個統計量值。",
        "en": "Every sample has exactly one statistic value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "sample=232",
      "after": "T(232)=0",
      "relation": "implies",
      "rule_id": "total_statistic_map",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個樣本點都恰有一個統計量值。",
        "en": "Every sample has exactly one statistic value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "sample=233",
      "after": "T(233)=1",
      "relation": "implies",
      "rule_id": "total_statistic_map",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個樣本點都恰有一個統計量值。",
        "en": "Every sample has exactly one statistic value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "[(230, 0), (231, 1), (232, 0), (233, 1)]",
      "after": "fibers=[[0, [230, 232]], [1, [231, 233]]]",
      "relation": "implies",
      "rule_id": "fiber_partition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依取值分組，全部fibers兩兩不交且聯集為完整樣本域。",
        "en": "Grouping by value produces disjoint fibers whose union is the complete carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        230,
        231,
        232,
        233
      ],
      [
        0,
        1,
        0,
        1
      ],
      [
        [
          0,
          [
            230,
            232
          ]
        ],
        [
          1,
          [
            231,
            233
          ]
        ]
      ],
      true,
      "total_statistic_fibers_partition_sample_space"
    ]
  },
  "conclusion": {
    "canonical_value": "([230, 231, 232, 233], [0, 1, 0, 1], [[0, [230, 232]], [1, [231, 233]]], 是, total_statistic_fibers_partition_sample_space)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "統計量是完整樣本集合上的全域映射；依取值分組的 fibers 互斥且覆蓋全部樣本。",
    "en": "The statistic is total on the sample carrier; the fibers by value are disjoint and cover every sample."
  },
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    "scope_note": {
      "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
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}