{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r04-conditional-expectation-0091",
  "pilot_number": "R0027-14191",
  "number": 14191,
  "family_id": "qf-mod29-condexp-total-expectation",
  "title": {
    "zh-Hant": "第 14191 題：有限條件期望",
    "en": "Question 14191: Total Expectation Is a Consequence, Not the Whole Definition"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r04-conditional-expectation-0091",
    "batch_id": "batch-20261014-advance-20261029-r04-conditional-expectation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-conditional-expectation"
    ],
    "title_zh": "有限條件期望",
    "stem_zh": "X機率law與conditioning partition綁同一有限空間。操作=condexp-total-expectation；資料={\"masses\":[[18,49],[24,49],[3,49],[4,49]],\"values\":[3,-2,17,-3],\"blocks\":[0,0,1,1]}。請依 [E_X,E_conditional,equal,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(45/49, 45/49, 是, tower_to_trivial_sigma_algebra)",
      "explanation_zh": "Wave4第三層核對=(45/49, 45/49, 是, tower_to_trivial_sigma_algebra)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；結果block可測並保持每個正質量block積分；零質量block只a.s.決定且不得除以零。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限條件期望"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-condexp-total-expectation",
      "seed": 202610294091,
      "parameters": {
        "masses": [
          [
            18,
            49
          ],
          [
            24,
            49
          ],
          [
            3,
            49
          ],
          [
            4,
            49
          ]
        ],
        "values": [
          3,
          -2,
          17,
          -3
        ],
        "blocks": [
          0,
          0,
          1,
          1
        ],
        "domain": "Exact_finite_partition_conditional_expectation",
        "operation": "condexp-total-expectation",
        "semantic_template": "qf-mod29-condexp-total-expectation/{\"blocks\":[0,0,1,1],\"masses\":[[18,49],[24,49],[3,49],[4,49]],\"values\":[3,-2,17,-3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261029-r04-conditional-expectation-0091",
    "batch_id": "batch-20261014-advance-20261029-r04-conditional-expectation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dc1d93743860aef53b4481673b015f8c6c70edfb54a500fe9cf1c88b59e307dc",
      "family_id": "qf-mod29-condexp-total-expectation",
      "seed": 202610294091,
      "semantic_template": "qf-mod29-condexp-total-expectation/{\"blocks\":[0,0,1,1],\"masses\":[[18,49],[24,49],[3,49],[4,49]],\"values\":[3,-2,17,-3]}"
    },
    "fields": {
      "title": "Total Expectation Is a Consequence, Not the Whole Definition",
      "stem": "Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).",
      "answer": {
        "kind": "expression",
        "value": "(45/49, 45/49, Yes, tower_to_trivial_sigma_algebra)",
        "explanation": "Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (45/49, 45/49, Yes, tower_to_trivial_sigma_algebra)."
      },
      "skills": [
        "Total Expectation Is a Consequence, Not the Whole Definition"
      ],
      "verification_warning": "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": "X機率law與conditioning partition綁同一有限空間。操作=condexp-total-expectation；資料={\"masses\":[[18,49],[24,49],[3,49],[4,49]],\"values\":[3,-2,17,-3],\"blocks\":[0,0,1,1]}。請依 [E_X,E_conditional,equal,certificate] 欄序回答。",
        "en": "Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "conditional model whole mass sum := sum(18/49, 24/49, 3/49, 4/49)",
      "after": "conditional model whole mass sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 conditional model whole mass sum，不以小數近似替代。",
          "en": "Recompute conditional model whole mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18/49",
          "24/49",
          "3/49",
          "4/49"
        ],
        "output": "1",
        "label": "conditional model whole mass sum"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "conditional model:raw aligned masses=[[18, 49], [24, 49], [3, 49], [4, 49]]",
      "after": "normalized masses=[18/49, 24/49, 3/49, 4/49];nonnegative=是;total=1",
      "relation": "implies",
      "rule_id": "whole_finite_probability_law_not_uniform_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整同一有限機率模型先核非負與總和恰為1；零質量原子仍保留，不默認均勻或另換模型。",
        "en": "First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "same outcome indices=[0, 1, 2, 3];X values=[3, -2, 17, -3];block labels=[0, 0, 1, 1]",
      "after": "distinct conditioning labels in increasing order=[0, 1]",
      "relation": "implies",
      "rule_id": "conditional_partition_same_space_original_indices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個標籤的一個纖維是一個conditioning區塊；標籤可不連續，輸出按數值標籤排序，但區塊內原子仍按原索引，不把區塊標籤當陣列位置。",
        "en": "Each label fiber is a conditioning block.Labels need not be consecutive.Sort summaries by label but retain original outcome indices within blocks;labels are not array positions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "block0 mass := sum(18/49, 24/49)",
      "after": "block0 mass = 6/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block0 mass，不以小數近似替代。",
          "en": "Recompute block0 mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18/49",
          "24/49"
        ],
        "output": "6/7",
        "label": "block0 mass"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "block0 atom0 weighted X := mul(18/49, 3)",
      "after": "block0 atom0 weighted X = 54/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block0 atom0 weighted X，不以小數近似替代。",
          "en": "Recompute block0 atom0 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18/49",
          "3"
        ],
        "output": "54/49",
        "label": "block0 atom0 weighted X"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "block0 atom1 weighted X := mul(24/49, -2)",
      "after": "block0 atom1 weighted X = -48/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block0 atom1 weighted X，不以小數近似替代。",
          "en": "Recompute block0 atom1 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24/49",
          "-2"
        ],
        "output": "-48/49",
        "label": "block0 atom1 weighted X"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "block0 original weighted integral := sum(54/49, -48/49)",
      "after": "block0 original weighted integral = 6/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block0 original weighted integral，不以小數近似替代。",
          "en": "Recompute block0 original weighted integral with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "54/49",
          "-48/49"
        ],
        "output": "6/49",
        "label": "block0 original weighted integral"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "block0 positive-mass conditional mean := div(6/49, 6/7)",
      "after": "block0 positive-mass conditional mean = 1/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block0 positive-mass conditional mean，不以小數近似替代。",
          "en": "Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6/49",
          "6/7"
        ],
        "output": "1/7",
        "label": "block0 positive-mass conditional mean"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "block label=0;indices=[0, 1];positive mass=6/7;weighted integral=6/49",
      "after": "block-constant conditional value=1/7",
      "relation": "implies",
      "rule_id": "positive_block_mean_including_zero_mean",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質量正才作加權積分除質量；平均等於0仍是已決定的數值0，不是零質量標記。把同一值複製到區塊每個原子。",
        "en": "Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "block1 mass := sum(3/49, 4/49)",
      "after": "block1 mass = 1/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block1 mass，不以小數近似替代。",
          "en": "Recompute block1 mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3/49",
          "4/49"
        ],
        "output": "1/7",
        "label": "block1 mass"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "block1 atom2 weighted X := mul(3/49, 17)",
      "after": "block1 atom2 weighted X = 51/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block1 atom2 weighted X，不以小數近似替代。",
          "en": "Recompute block1 atom2 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/49",
          "17"
        ],
        "output": "51/49",
        "label": "block1 atom2 weighted X"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "block1 atom3 weighted X := mul(4/49, -3)",
      "after": "block1 atom3 weighted X = -12/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block1 atom3 weighted X，不以小數近似替代。",
          "en": "Recompute block1 atom3 weighted X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/49",
          "-3"
        ],
        "output": "-12/49",
        "label": "block1 atom3 weighted X"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "block1 original weighted integral := sum(51/49, -12/49)",
      "after": "block1 original weighted integral = 39/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block1 original weighted integral，不以小數近似替代。",
          "en": "Recompute block1 original weighted integral with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "51/49",
          "-12/49"
        ],
        "output": "39/49",
        "label": "block1 original weighted integral"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "block1 positive-mass conditional mean := div(39/49, 1/7)",
      "after": "block1 positive-mass conditional mean = 39/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 block1 positive-mass conditional mean，不以小數近似替代。",
          "en": "Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "39/49",
          "1/7"
        ],
        "output": "39/7",
        "label": "block1 positive-mass conditional mean"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "block label=1;indices=[2, 3];positive mass=1/7;weighted integral=39/49",
      "after": "block-constant conditional value=39/7",
      "relation": "implies",
      "rule_id": "positive_block_mean_including_zero_mean",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質量正才作加權積分除質量；平均等於0仍是已決定的數值0，不是零質量標記。把同一值複製到區塊每個原子。",
        "en": "Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "total original expectation := sum(6/49, 39/49)",
      "after": "total original expectation = 45/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total original expectation，不以小數近似替代。",
          "en": "Recompute total original expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6/49",
          "39/49"
        ],
        "output": "45/49",
        "label": "total original expectation"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "total expectation block0 conditional integral := mul(6/7, 1/7)",
      "after": "total expectation block0 conditional integral = 6/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total expectation block0 conditional integral，不以小數近似替代。",
          "en": "Recompute total expectation block0 conditional integral with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6/7",
          "1/7"
        ],
        "output": "6/49",
        "label": "total expectation block0 conditional integral"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "total expectation block1 conditional integral := mul(1/7, 39/7)",
      "after": "total expectation block1 conditional integral = 39/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total expectation block1 conditional integral，不以小數近似替代。",
          "en": "Recompute total expectation block1 conditional integral with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/7",
          "39/7"
        ],
        "output": "39/49",
        "label": "total expectation block1 conditional integral"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "total conditional expectation := sum(6/49, 39/49)",
      "after": "total conditional expectation = 45/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total conditional expectation，不以小數近似替代。",
          "en": "Recompute total conditional expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6/49",
          "39/49"
        ],
        "output": "45/49",
        "label": "total conditional expectation"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "all conditioning blocks partition the same carrier;E_X=45/49;E_conditional=45/49",
      "after": "equal=是;tower_to_trivial_sigma_algebra",
      "relation": "implies",
      "rule_id": "total_expectation_consequence_not_whole_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把每個區塊積分相等加總才得到總期望相等；這是到平凡sigma代數的tower後果，不反向用單一總和取代可測性及全部區塊義務。",
        "en": "Sum the blockwise integral identities to obtain total expectation equality.This is the tower consequence to the trivial sigma-algebra,not a reverse substitution for measurability and every block obligation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "45/49",
      "45/49",
      true,
      "tower_to_trivial_sigma_algebra"
    ]
  },
  "conclusion": {
    "canonical_value": "(45/49, 45/49, 是, tower_to_trivial_sigma_algebra)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261029-r04-conditional-expectation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r04-conditional-expectation-foundations/questions.jsonl",
      "batch_sha256": "f10ecb9a7685a673bd8efa3ee909031091d6faa4ad4ac4afe976e1bd82fe4eb6",
      "original_line": 91,
      "original_line_sha256": "79ad570d81ec4290fb96baa3ef52ca9264913db6a3895fab353ba66c4f1d41fe",
      "question_compact_sha256": "dc1d93743860aef53b4481673b015f8c6c70edfb54a500fe9cf1c88b59e307dc",
      "english_row_compact_sha256": "6f2cdcfcfce7c9255c769772202929f8ea857d4d2be2cf91693699fbfa085def",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r04-conditional-expectation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r04-conditional-expectation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-conditional-expectation": {
        "sha256": "35d7f871dac286e45d4661dcffbafe25a02cca5c55a4c58ef60bd41d2408bd95",
        "english_sha256": "859840df1ea5def7705f723675b947d841b733e899bda95980bcd58b6f62644c",
        "original_path": "public/data/mko/conditional-expectation.json",
        "english_path": "public/data/i18n/editions/mko-conditional-expectation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77",
    "original_candidate_sha256": "1ec1387628d62674e3109faad6fc66ec854c47638ad1ec7e3d67bb459b447b27",
    "original_candidate_line_sha256": "1ec1387628d62674e3109faad6fc66ec854c47638ad1ec7e3d67bb459b447b27"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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."
  },
  "revision": {
    "id": "review-20261026-007-p28",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe",
      "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "02eb4590d7abb2b77e651cd97764032a0e50df3c62cd11aa63f34d336f9743b1",
      "arithmetic_steps": 15
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}