{
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  "question_id": "q-20261014-advance-20261031-r03-convergence-in-distribution-0068",
  "pilot_number": "R0030-15743",
  "number": 15743,
  "family_id": "qf-mod31-convdist-continuity-obligation",
  "title": {
    "zh-Hant": "第 15743 題：依分布收斂",
    "en": "Question 15743: Distinguish Required Continuity Points from Limit-CDF Jumps"
  },
  "original_question": {
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    "id": "q-20261014-advance-20261031-r03-convergence-in-distribution-0068",
    "batch_id": "batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-convergence-in-distribution"
    ],
    "title_zh": "依分布收斂",
    "stem_zh": "Bernoulli symbolic tail與極限CDF。操作=convdist-continuity-obligation；資料={\"p\":[18,30],\"threshold\":[5,7]}。請依 [p,threshold,continuity_point,definition_obligation] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3/5, 5/7, 是, CDF_convergence_required_here)",
      "explanation_zh": "Wave4收束核對=(3/5, 5/7, 是, CDF_convergence_required_here)；依完整明示資料重算；只在極限CDF連續點要求收斂且有限prefix不足；不要求coupling且不推出依機率、幾乎處處或Lp收斂。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "依分布收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-convdist-continuity-obligation",
      "seed": 202610313068,
      "parameters": {
        "p": [
          18,
          30
        ],
        "threshold": [
          5,
          7
        ],
        "domain": "Symbolic_Bernoulli_CDF_limit",
        "operation": "convdist-continuity-obligation",
        "semantic_template": "qf-mod31-convdist-continuity-obligation/{\"p\":[18,30],\"threshold\":[5,7]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
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        "producer_substantive_template_deduplication",
        "producer_json_schema"
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      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
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      "generated_at": "2026-10-14",
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    "question_id": "q-20261014-advance-20261031-r03-convergence-in-distribution-0068",
    "batch_id": "batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations",
    "locale": "en",
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      "family_id": "qf-mod31-convdist-continuity-obligation",
      "seed": 202610313068,
      "semantic_template": "qf-mod31-convdist-continuity-obligation/{\"p\":[18,30],\"threshold\":[5,7]}"
    },
    "fields": {
      "title": "Distinguish Required Continuity Points from Limit-CDF Jumps",
      "stem": "Use the explicit Bernoulli symbolic family with interior limit p=18/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.",
      "answer": {
        "kind": "expression",
        "value": "(3/5, 5/7, Yes, CDF_convergence_required_here)",
        "explanation": "For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/5, 5/7, Yes, CDF_convergence_required_here)."
      },
      "skills": [
        "Distinguish Required Continuity Points from Limit-CDF Jumps"
      ],
      "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": "Bernoulli symbolic tail與極限CDF。操作=convdist-continuity-obligation；資料={\"p\":[18,30],\"threshold\":[5,7]}。請依 [p,threshold,continuity_point,definition_obligation] 欄序回答。",
        "en": "Use the explicit Bernoulli symbolic family with interior limit p=18/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "interior Bernoulli limit mass at0 := sub(1, 3/5)",
      "after": "interior Bernoulli limit mass at0 = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 interior Bernoulli limit mass at0，不以小數近似替代。",
          "en": "Recompute interior Bernoulli limit mass at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "3/5"
        ],
        "output": "2/5",
        "label": "interior Bernoulli limit mass at0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original limit parameter=[18, 30];reduced p=3/5;mass at0=2/5;mass at1=3/5",
      "after": "both endpoint masses positive;only limit-CDF jumps at0 and1;registered interior projection",
      "relation": "implies",
      "rule_id": "Bernoulli_interior_English_guard_not_degenerate_jump_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先遵守原英文0<p<1限制；因此0與1的跳躍都真有正質量。原始oracle雖允許端點p，也不把退化分佈套進這個連續點協定；它們須另行分析。",
        "en": "Honor the source English restriction0<p<1,so both jumps genuinely have positive mass.Although the raw oracle allows endpoint parameters,do not extend this continuity-point protocol to degenerate distributions;their jump sets require separate analysis."
      },
      "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": "interior limit p=3/5;exact threshold=5/7;jump set=[0, 1]",
      "after": "limit CDF continuity point=是;definition obligation=CDF_convergence_required_here",
      "relation": "implies",
      "rule_id": "convdist_limit_continuity_obligation_not_convergence_failure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依分佈收斂只要求極限CDF的每個連續點；跳點是定義不要求，不是斷言該點一定不收斂，也不是看一個n判連續。",
        "en": "Convergence in distribution requires convergence at every continuity point of the limit CDF.A jump point is not required by the definition;this is not a claim of failure there,and continuity is not decided from one n."
      },
      "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": [
      "3/5",
      "5/7",
      true,
      "CDF_convergence_required_here"
    ]
  },
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    "canonical_value": "(3/5, 5/7, 是, CDF_convergence_required_here)",
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      "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."
    }
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    "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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