{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r05-expectation-0012",
  "pilot_number": "R0007-04112",
  "number": 4112,
  "family_id": "qf-density10-expectation-contributions",
  "title": {
    "zh-Hant": "第 4112 題：逐點期望貢獻",
    "en": "Question 4112: Every Weighted Expectation Contribution"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r05-expectation-0012",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-expectation"
    ],
    "title_zh": "逐點期望貢獻",
    "stem_zh": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。",
    "answer": {
      "kind": "expression",
      "value": "[-44/5, 4/15, 1/15]",
      "explanation_zh": "逐點期望項=[-44/5, 4/15, 1/15]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "逐點期望貢獻"
      ]
    },
    "variant": {
      "family_id": "qf-density10-expectation-contributions",
      "seed": 202610105012,
      "parameters": {
        "masses": [
          12,
          2,
          1
        ],
        "D": 15,
        "values": [
          -11,
          2,
          1
        ],
        "operation": "density10_expectation_contributions",
        "domain": "finite_integer_random_variable",
        "semantic_template": "qf-density10-expectation-contributions/[[-11,\"4/5\"],[2,\"2/15\"],[1,\"1/15\"]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r05-expectation-0012",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dcda0b11de1781aa712eb8276b73b4274196afe3908679b89f2b21be27217b6b",
      "family_id": "qf-density10-expectation-contributions",
      "seed": 202610105012,
      "semantic_template": "qf-density10-expectation-contributions/[[-11,\"4/5\"],[2,\"2/15\"],[1,\"1/15\"]]"
    },
    "fields": {
      "title": "Every Weighted Expectation Contribution",
      "stem": "S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "[-44/5, 4/15, 1/15]",
        "explanation": "Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-44/5, 4/15, 1/15]."
      },
      "skills": [
        "Every Weighted Expectation Contribution"
      ],
      "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": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。",
        "en": "S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "total mass numerator := sum(12, 2, 1)",
      "after": "total mass numerator = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負整數質量要加總成D，權重才恰歸一化。",
        "en": "Nonnegative integer masses must sum to D for normalized weights."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total mass numerator，不以小數近似替代。",
          "en": "Recompute total mass numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "2",
          "1"
        ],
        "output": "15",
        "label": "total mass numerator"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "indexed masses=[12, 2, 1];D=15",
      "after": "probabilities=[4/5, 2/15, 1/15]",
      "relation": "implies",
      "rule_id": "indexed_normalized_expectation_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本索引從0起，質量、X與Y值按相同索引對齊；零質量結果仍保留，負隨機變數值合法。",
        "en": "Outcome indices start at0 and align masses with X and Y values.Retain zero-mass outcomes;negative random-variable values are allowed."
      },
      "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": "X weighted numerator0 := mul(12, -11)",
      "after": "X weighted numerator0 = -132",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator0，不以小數近似替代。",
          "en": "Recompute X weighted numerator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "-11"
        ],
        "output": "-132",
        "label": "X weighted numerator0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "X contribution0 := div(-132, 15)",
      "after": "X contribution0 = -44/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution0，不以小數近似替代。",
          "en": "Recompute X contribution0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-132",
          "15"
        ],
        "output": "-44/5",
        "label": "X contribution0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "X weighted numerator1 := mul(2, 2)",
      "after": "X weighted numerator1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator1，不以小數近似替代。",
          "en": "Recompute X weighted numerator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "X weighted numerator1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "X contribution1 := div(4, 15)",
      "after": "X contribution1 = 4/15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution1，不以小數近似替代。",
          "en": "Recompute X contribution1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "15"
        ],
        "output": "4/15",
        "label": "X contribution1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "X weighted numerator2 := mul(1, 1)",
      "after": "X weighted numerator2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將此索引質量乘同索引值，不能刪掉零質量後錯位。",
        "en": "Multiply the mass by the value at the same index;deleting zero masses must not shift alignment."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X weighted numerator2，不以小數近似替代。",
          "en": "Recompute X weighted numerator2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "X weighted numerator2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "X contribution2 := div(1, 15)",
      "after": "X contribution2 = 1/15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母；每個零項也保留其索引位置。",
        "en": "Divide by the common positive denominator,retaining the index position of a zero contribution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 X contribution2，不以小數近似替代。",
          "en": "Recompute X contribution2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "15"
        ],
        "output": "1/15",
        "label": "X contribution2"
      }
    }
  ],
  "math_model": {
    "result": [
      "-44/5",
      "4/15",
      "1/15"
    ],
    "probabilities": [
      "4/5",
      "2/15",
      "1/15"
    ]
  },
  "conclusion": {
    "canonical_value": "[-44/5, 4/15, 1/15]",
    "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-20261008-advance-20261010-r05-expectation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r05-expectation-foundations/questions.jsonl",
      "batch_sha256": "4ff64bd7b738bff274c56f73708b9491e71e052b39893de82b63de7618bd6fa2",
      "original_line": 12,
      "original_line_sha256": "6e04223de6d43fb1ba19d9de3982765a3145dfa46c0e556a285f90de097add61",
      "question_compact_sha256": "dcda0b11de1781aa712eb8276b73b4274196afe3908679b89f2b21be27217b6b",
      "english_row_compact_sha256": "703154438c55766206295406ae4b5a766ba8827aa965543c1fc232cce8054f57",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r05-expectation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r05-expectation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-expectation": {
        "sha256": "71bc68faf69ca0df9cdd11d093055262aa277d07abd77982d0aa1afd849daf3c",
        "english_sha256": "636015fa1e3a6d9eae6091042ae3dd70c6cb996ecd9a409c890060a5b503333a",
        "original_path": "public/data/mko/expectation.json",
        "english_path": "public/data/i18n/editions/mko-expectation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "63077c57fca41c6183cde71429c0f54ae79f693aeb47d633ec5ec2379898ef76",
    "original_candidate_line_sha256": "63077c57fca41c6183cde71429c0f54ae79f693aeb47d633ec5ec2379898ef76"
  },
  "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-20261026-004-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0d2012f03dff8cc1ab380ca58e0d31bf1444a05aeae8fa4d32ed670fa149b009",
      "arithmetic_steps": 7
    },
    "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."
  }
}