{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r05-expectation-0077",
  "pilot_number": "R0007-04152",
  "number": 4152,
  "family_id": "qf-density10-expectation-missing-value",
  "title": {
    "zh-Hant": "第 4152 題：給定均值的未知整數值",
    "en": "Question 4152: Positive-Mass Value Recovered from a Mean"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r05-expectation-0077",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-expectation"
    ],
    "title_zh": "給定均值的未知整數值",
    "stem_zh": "有限歸一化模型；S={0,…,0}，F=𝒫(S)，質量=[1]、D=1，權重為質量/D；X值=[null]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=0/1。求該槽整數值，仍寫成p/1。",
    "answer": {
      "kind": "expression",
      "value": "0/1",
      "explanation_zh": "未知整數值=0/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "給定均值的未知整數值"
      ]
    },
    "variant": {
      "family_id": "qf-density10-expectation-missing-value",
      "seed": 202610105077,
      "parameters": {
        "masses": [
          1
        ],
        "D": 1,
        "values": [
          null
        ],
        "k": 0,
        "mean_n": 0,
        "mean_d": 1,
        "operation": "density10_expectation_missing_value",
        "domain": "finite_integer_random_variable",
        "semantic_template": "qf-density10-expectation-missing-value/[0,[null],\"1/1\",\"0/1\"]"
      }
    },
    "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-0077",
    "batch_id": "batch-20261008-advance-20261010-r05-expectation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f6af40853c660df938335ad5569c92d06d5a5618a0f71d132118be43f6ff40b7",
      "family_id": "qf-density10-expectation-missing-value",
      "seed": 202610105077,
      "semantic_template": "qf-density10-expectation-missing-value/[0,[null],\"1/1\",\"0/1\"]"
    },
    "fields": {
      "title": "Positive-Mass Value Recovered from a Mean",
      "stem": "S={0,…,0}, F=P(S), masses=[1], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=0/1. Find that slot’s integer value, written as p/1.",
      "answer": {
        "kind": "expression",
        "value": "0/1",
        "explanation": "Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 0/1."
      },
      "skills": [
        "Positive-Mass Value Recovered from a Mean"
      ],
      "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,…,0}，F=𝒫(S)，質量=[1]、D=1，權重為質量/D；X值=[null]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=0/1。求該槽整數值，仍寫成p/1。",
        "en": "S={0,…,0}, F=P(S), masses=[1], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=0/1. Find that slot’s integer value, written as p/1."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "total mass numerator := sum(1)",
      "after": "total mass numerator = 1",
      "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": [
          "1"
        ],
        "output": "1",
        "label": "total mass numerator"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "indexed masses=[1];D=1",
      "after": "probabilities=[1/1]",
      "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": "known weighted numerator := sum()",
      "after": "known weighted numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將全部已知質量值項相加。",
        "en": "Add all known mass-value terms."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 known weighted numerator，不以小數近似替代。",
          "en": "Recompute known weighted numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [],
        "output": "0",
        "label": "known weighted numerator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "target expectation numerator := mul(0, 1)",
      "after": "target expectation numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把目標均值乘D，得到全部加權分子的目標。",
        "en": "Multiply the target mean by D to obtain the target weighted numerator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 target expectation numerator，不以小數近似替代。",
          "en": "Recompute target expectation numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "target expectation numerator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "unknown weighted contribution := sub(0, 0)",
      "after": "unknown weighted contribution = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "扣除已知項，剩m_k·未知值。",
        "en": "Subtract known terms to leave mass_k times the unknown value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 unknown weighted contribution，不以小數近似替代。",
          "en": "Recompute unknown weighted contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "unknown weighted contribution"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "recovered unknown value := div(0, 1)",
      "after": "recovered unknown value = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有未知槽質量正才可唯一反解；此題切片要求值為整數。",
        "en": "Positive unknown-slot mass makes the value uniquely recoverable;this exercise slice requires an integer result."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recovered unknown value，不以小數近似替代。",
          "en": "Recompute recovered unknown value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "recovered unknown value"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "recovered weighted numerator := mul(1, 0)",
      "after": "recovered weighted numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用恢復值乘未知槽原質量，反核對齊及符號。",
        "en": "Multiply the recovered value by its original slot mass to audit alignment and sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recovered weighted numerator，不以小數近似替代。",
          "en": "Recompute recovered weighted numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "recovered weighted numerator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "complete reconstructed numerator := sum(0, 0)",
      "after": "complete reconstructed numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "補回未知項得到完整分子。",
        "en": "Restore the missing term to obtain the complete numerator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 complete reconstructed numerator，不以小數近似替代。",
          "en": "Recompute complete reconstructed numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "complete reconstructed numerator"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "reconstructed expectation := div(0, 1)",
      "after": "reconstructed expectation = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以D反代，必須等於明示均值。",
        "en": "Divide by D to substitute back;the result must equal the stated mean."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reconstructed expectation，不以小數近似替代。",
          "en": "Recompute reconstructed expectation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "reconstructed expectation"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "k=0;mass_k=1>0;target=0",
      "after": "unique recovered value=0;substitution mean=0",
      "relation": "implies",
      "rule_id": "positive_mass_missing_value_uniqueness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若質量0，均值不限制這個槽，不能沿用本反解；null是唯一未知標記，不是數值0。",
        "en": "With mass0 the mean would not constrain this slot,so this inversion would fail.Null marks the unique unknown and is not numeric0."
      },
      "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": "0",
    "known_weighted_numerator": "0",
    "target_mean": "0",
    "recovered_mean": "0"
  },
  "conclusion": {
    "canonical_value": "0/1",
    "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": 77,
      "original_line_sha256": "774ef4c7e4d6ba6218b7be293d3dfb1404250b215e8bbad175a1a00226332c16",
      "question_compact_sha256": "f6af40853c660df938335ad5569c92d06d5a5618a0f71d132118be43f6ff40b7",
      "english_row_compact_sha256": "d984005a68ddb8db84576044d55b1a2ed7e80502cd34dbeeb6bdbdd1bc08c905",
      "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": {
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        "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": "48d3e005da5c1cee3baa21ca0a52491c0b3aaa9ae719f20a6d3d6ef67c977370",
    "original_candidate_line_sha256": "48d3e005da5c1cee3baa21ca0a52491c0b3aaa9ae719f20a6d3d6ef67c977370"
  },
  "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": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "31f366c98e759d23ed008c0919ca543fcb45c44ed2ca0ca3086ce9e17e00f83b",
      "arithmetic_steps": 8
    },
    "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."
  }
}