{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261030-r06-cumulative-distribution-function-0013",
  "pilot_number": "R0028-14913",
  "number": 14913,
  "family_id": "qf-mod30-cdf-evaluate",
  "title": {
    "zh-Hant": "第 14913 題：累積分布函數",
    "en": "Question 14913: The CDF Includes Mass Exactly at the Threshold"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261030-r06-cumulative-distribution-function-0013",
    "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-evaluate；資料={\"support\":[0,7,13],\"masses\":[[1,4],[1,4],[1,2]],\"t\":0}。請依 [t,F_t,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 1/4, uses_X_less_than_or_equal_t)",
      "explanation_zh": "Wave4第四層核對=(0, 1/4, uses_X_less_than_or_equal_t)；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-evaluate",
      "seed": 202610306013,
      "parameters": {
        "support": [
          0,
          7,
          13
        ],
        "masses": [
          [
            1,
            4
          ],
          [
            1,
            4
          ],
          [
            1,
            2
          ]
        ],
        "t": 0,
        "domain": "Exact_finite_discrete_CDF",
        "operation": "cdf-evaluate",
        "semantic_template": "qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[0,7,13],\"t\":0}"
      }
    },
    "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-0013",
    "batch_id": "batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3a3accdb8e93b2166166c1f7a56667f95374c7bddd3bbc946ee3c7fecb72876e",
      "family_id": "qf-mod30-cdf-evaluate",
      "seed": 202610306013,
      "semantic_template": "qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[0,7,13],\"t\":0}"
    },
    "fields": {
      "title": "The CDF Includes Mass Exactly at the Threshold",
      "stem": "Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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. Threshold t=0. Give (t,F_t,uses_X_less_than_or_equal_t).",
      "answer": {
        "kind": "expression",
        "value": "(0, 1/4, uses_X_less_than_or_equal_t)",
        "explanation": "Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (0, 1/4, uses_X_less_than_or_equal_t)."
      },
      "skills": [
        "The CDF Includes Mass Exactly at the Threshold"
      ],
      "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-evaluate；資料={\"support\":[0,7,13],\"masses\":[[1,4],[1,4],[1,2]],\"t\":0}。請依 [t,F_t,boundary] 欄序回答。",
        "en": "Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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. Threshold t=0. Give (t,F_t,uses_X_less_than_or_equal_t)."
      }
    }
  ],
  "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=[0, 7, 13];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": "threshold CDF:original support index=0;value=0;threshold=0",
      "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": "threshold CDF:original support index=1;value=7;threshold=0",
      "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": "threshold CDF:original support index=2;value=13;threshold=0",
      "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": "threshold CDF selected mass sum := sum(1/4)",
      "after": "threshold 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 threshold CDF selected mass sum，不以小數近似替代。",
          "en": "Recompute threshold CDF selected mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4"
        ],
        "output": "1/4",
        "label": "threshold CDF selected mass sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "threshold=0;cumulative mass=1/4",
      "after": "F(t)=P(X<=t),not point mass or density;finite-law tails0 and1",
      "relation": "implies",
      "rule_id": "CDF_inclusive_value_and_endpoint_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "低於全部支撐時空和0，達最大支撐後總質量1；F(t)加總所有≤t的原子，不是只取X=t的PMF，也不是density。",
        "en": "Below all support the empty sum is0;at/above the maximum the total is1.F(t) sums every atom≤t,not only the point mass P(X=t) and not a density."
      },
      "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,
      "1/4",
      "uses_X_less_than_or_equal_t"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 1/4, uses_X_less_than_or_equal_t)",
    "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."
  },
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        "original_path": "public/data/mko/cumulative-distribution-function.json",
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_sha256": "e4e59e846aca886d1cc0a20daf31c3bed34affd4b0fb61e2f89e920c40432a3e",
    "original_candidate_line_sha256": "e4e59e846aca886d1cc0a20daf31c3bed34affd4b0fb61e2f89e920c40432a3e"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "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-p29",
    "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,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}