{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261030-r06-cumulative-distribution-function-0066",
  "pilot_number": "R0028-14966",
  "number": 14966,
  "family_id": "qf-mod30-cdf-monotonicity",
  "title": {
    "zh-Hant": "第 14966 題：累積分布函數",
    "en": "Question 14966: Nested Threshold Events Give a Nondecreasing CDF"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261030-r06-cumulative-distribution-function-0066",
    "batch_id": "batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-cumulative-distribution-function"
    ],
    "title_zh": "累積分布函數",
    "stem_zh": "完整有限離散law與F(t)=P(X小於等於t)。操作=cdf-monotonicity；資料={\"support\":[3,10,16],\"masses\":[[1,4],[1,4],[1,2]],\"left\":3,\"right\":16}。請依 [left,right,F_left,F_right,monotone] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3, 16, 1/4, 1, 是)",
      "explanation_zh": "Wave4第四層核對=(3, 16, 1/4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；保留小於等於單調右連續與端點；CDF不混同PMF或density且jump為點質量。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "累積分布函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod30-cdf-monotonicity",
      "seed": 202610306066,
      "parameters": {
        "support": [
          3,
          10,
          16
        ],
        "masses": [
          [
            1,
            4
          ],
          [
            1,
            4
          ],
          [
            1,
            2
          ]
        ],
        "left": 3,
        "right": 16,
        "domain": "Exact_finite_discrete_CDF",
        "operation": "cdf-monotonicity",
        "semantic_template": "qf-mod30-cdf-monotonicity/{\"left\":3,\"masses\":[[1,4],[1,4],[1,2]],\"right\":16,\"support\":[3,10,16]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_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、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或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-20261030-r06-cumulative-distribution-function-0066",
    "batch_id": "batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ee4f0dd3b796139dcff3e6e1eff43c8dca42baea815df9abd7304f95f1d88d60",
      "family_id": "qf-mod30-cdf-monotonicity",
      "seed": 202610306066,
      "semantic_template": "qf-mod30-cdf-monotonicity/{\"left\":3,\"masses\":[[1,4],[1,4],[1,2]],\"right\":16,\"support\":[3,10,16]}"
    },
    "fields": {
      "title": "Nested Threshold Events Give a Nondecreasing CDF",
      "stem": "Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=3≤right=16. Give (left,right,F_left,F_right,monotone?).",
      "answer": {
        "kind": "expression",
        "value": "(3, 16, 1/4, 1, Yes)",
        "explanation": "Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (3, 16, 1/4, 1, Yes)."
      },
      "skills": [
        "Nested Threshold Events Give a Nondecreasing CDF"
      ],
      "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": "完整有限離散law與F(t)=P(X小於等於t)。操作=cdf-monotonicity；資料={\"support\":[3,10,16],\"masses\":[[1,4],[1,4],[1,2]],\"left\":3,\"right\":16}。請依 [left,right,F_left,F_right,monotone] 欄序回答。",
        "en": "Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=3≤right=16. Give (left,right,F_left,F_right,monotone?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "CDF law total mass := sum(1/4, 1/4, 1/2)",
      "after": "CDF law total mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CDF law total mass，不以小數近似替代。",
          "en": "Recompute CDF law total mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/4",
          "1/2"
        ],
        "output": "1",
        "label": "CDF law total mass"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original ordered support=[3, 10, 16];raw aligned masses=[[1, 4], [1, 4], [1, 2]]",
      "after": "normalized aligned masses=[1/4, 1/4, 1/2];nonnegative=是;total=1",
      "relation": "implies",
      "rule_id": "CDF_complete_probability_law_original_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整離散law先核支撐互異、質量非負且總和1；即使支撐未排序也保留原配對，零質量支撐點不刪掉。這裡獨立守住來源英文宣告的機率law前提。",
        "en": "Qualify distinct support,nonnegative masses and unit total.Preserve original support-mass pairing even if unsorted,and retain zero-mass support points.This independently enforces the probability-law premise declared by the source English scope."
      },
      "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": "left CDF:original support index=0;value=3;threshold=3",
      "after": "test value<=threshold=是;aligned mass=1/4",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "left CDF:original support index=1;value=10;threshold=3",
      "after": "test value<=threshold=否;aligned mass=1/4",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "left CDF:original support index=2;value=16;threshold=3",
      "after": "test value<=threshold=否;aligned mass=1/2",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "left CDF selected mass sum := sum(1/4)",
      "after": "left CDF selected mass sum = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left CDF selected mass sum，不以小數近似替代。",
          "en": "Recompute left CDF selected mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4"
        ],
        "output": "1/4",
        "label": "left CDF selected mass sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "right CDF:original support index=0;value=3;threshold=16",
      "after": "test value<=threshold=是;aligned mass=1/4",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "right CDF:original support index=1;value=10;threshold=16",
      "after": "test value<=threshold=是;aligned mass=1/4",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "right CDF:original support index=2;value=16;threshold=16",
      "after": "test value<=threshold=是;aligned mass=1/2",
      "relation": "implies",
      "rule_id": "CDF_every_original_atom_threshold_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原支撐索引核條件再取該原子質量，分清嚴格<與含等號≤；不能排序支撐後仍沿用舊質量位置。",
        "en": "Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "right CDF selected mass sum := sum(1/4, 1/4, 1/2)",
      "after": "right CDF selected mass sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right CDF selected mass sum，不以小數近似替代。",
          "en": "Recompute right CDF selected mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/4",
          "1/2"
        ],
        "output": "1",
        "label": "right CDF selected mass sum"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "CDF interval left-open right-closed mass := sum(1/4, 1/2)",
      "after": "CDF interval left-open right-closed mass = 3/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CDF interval left-open right-closed mass，不以小數近似替代。",
          "en": "Recompute CDF interval left-open right-closed mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/2"
        ],
        "output": "3/4",
        "label": "CDF interval left-open right-closed mass"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "CDF monotone signed difference := sub(1, 1/4)",
      "after": "CDF monotone signed difference = 3/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CDF monotone signed difference，不以小數近似替代。",
          "en": "Recompute CDF monotone signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1/4"
        ],
        "output": "3/4",
        "label": "CDF monotone signed difference"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ordered thresholds=[3, 16];F_left=1/4;F_right=1;difference=3/4",
      "after": "threshold events nested;extra interval mass>=0;nondecreasing=是",
      "relation": "implies",
      "rule_id": "CDF_monotonicity_nested_events_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "{X≤left}包含於{X≤right}，差是(left,right]事件的非負質量，對任意有序門檻成立，不從兩個數值外推CDF公理。",
        "en": "The left threshold event is contained in the right one.The difference is the nonnegative mass of(left,right],for arbitrary ordered thresholds,not an empirical extrapolation from two values."
      },
      "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": [
      3,
      16,
      "1/4",
      "1",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 16, 1/4, 1, 是)",
    "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-20261030-r06-cumulative-distribution-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations/questions.jsonl",
      "batch_sha256": "9e9ea5de9610598dde1afc9f71b6840d412785c507b7430ba77426f45ea6fefa",
      "original_line": 66,
      "original_line_sha256": "d9e311ae3631ba40c216f8121123fe3162fdc78cfe6e8b49906f9854a185cddb",
      "question_compact_sha256": "ee4f0dd3b796139dcff3e6e1eff43c8dca42baea815df9abd7304f95f1d88d60",
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      "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."
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    "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": {
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    "bilingual_semantics": true,
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}