{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261111-r04-continuous-real-distribution-function-0088",
  "pilot_number": "R0044-22863",
  "number": 22863,
  "family_id": "qf-mod42-contcdf-singular-no-density-boundary",
  "title": {
    "zh-Hant": "第 22863 題：連續實值分布函數",
    "en": "Question 22863: Continuous and Atomless Does Not Imply a Density"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261111-r04-continuous-real-distribution-function-0088",
    "batch_id": "batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-continuous-real-distribution-function"
    ],
    "title_zh": "連續實值分布函數",
    "stem_zh": "CDF本身continuous、nondecreasing、right-continuous且audit query_points嚴格遞增。操作=contcdf-singular-no-density-boundary；資料={\"left\":-13,\"right\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-13, 14, 是, 是, 否, 是, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)",
      "explanation_zh": "Wave6第五層核對=(-13, 14, 是, 是, 否, 是, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)；依完整明示資料獨立重算；continuous CDF無jumps所以singletons質量0，flat intervals仍允許且兩端limits為0與1；continuous或atomless不推出density或absolute continuity；singular Cantor law是boundary。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "連續實值分布函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod42-contcdf-singular-no-density-boundary",
      "seed": 202611114088,
      "parameters": {
        "left": -13,
        "right": 14,
        "domain": "Exact_continuous_piecewise_linear_CDF",
        "operation": "contcdf-singular-no-density-boundary",
        "semantic_template": "qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-13,\"right\":14}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r04-continuous-real-distribution-function-0088",
    "batch_id": "batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "232bf805d8f3e535556b90e04dfa1b8f82327146c3ad343984072db1d8a6b360",
      "family_id": "qf-mod42-contcdf-singular-no-density-boundary",
      "seed": 202611114088,
      "semantic_template": "qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-13,\"right\":14}"
    },
    "fields": {
      "title": "Continuous and Atomless Does Not Imply a Density",
      "stem": "Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)",
        "explanation": "Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."
      },
      "skills": [
        "Continuous and Atomless Does Not Imply a Density"
      ],
      "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": "CDF本身continuous、nondecreasing、right-continuous且audit query_points嚴格遞增。操作=contcdf-singular-no-density-boundary；資料={\"left\":-13,\"right\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=contcdf-singular-no-density-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[left, right, CDF_continuous, atomless, has_density, singular, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。",
        "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness."
      },
      "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": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Cantor affine support positive width := sub(14, -13)",
      "after": "Cantor affine support positive width = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Cantor affine support positive width，不以小數近似替代。",
          "en": "Recompute Cantor affine support positive width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "-13"
        ],
        "output": "27",
        "label": "Cantor affine support positive width"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL affine Cantor support[-13,14];positive width W=27",
      "after": "take independent fair digits B_k in{0,1},Y=sum_(k>=1)2*B_k/3^k and X=L+W*Y;the geometric tail converges and X lies in the affine Cantor support",
      "relation": "implies",
      "rule_id": "contcdf_Cantor_WHOLE_binary_digit_construction_not_knots",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是完整指定的Cantor law，不呼叫piecewise-linear建構或拿有限CDF樣本冒充證據。0/2三進位級數給機率1的Cantor支集，正仿射變換保非退化。",
        "en": "This defines the whole Cantor law,not the piecewise-linear constructor or a finite CDF sample.The convergent ternary0/2 series has probability1 on Cantor support,and the positive affine transformation is nondegenerate."
      },
      "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": "ORIGINAL support width W=27;level n ternary construction uses2^n intervals of length W/3^n",
      "after": "total level-n length=W*(2/3)^n tends0,so support has Lebesgue measure0;any singleton has probability<=2*2^(-n) for EVERY n,hence atom probability0;CDF jumps are0",
      "relation": "implies",
      "rule_id": "contcdf_Cantor_null_support_and_vanishing_atom_bounds",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各層包含支集且總長度趨0，證Lebesgue零測。固定點至多落在相應的有限個cylinder，每個前n位指定機率2^−n；對所有n的趨零上界才證無atom，並非有限點取樣。",
        "en": "The nested level sets contain the support and their total length tends0,proving Lebesgue-null support.A fixed point is bounded by at most two relevant cylinder masses2^(-n).The bound for every n proves no atoms,not sampling finitely many points."
      },
      "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": "ORIGINAL affine law on[-13,14] has support N with Lebesgue measure0 and probability1;all singleton probabilities0",
      "after": "CDF_continuous=是;atomless=是;has_density=否;singular=是;any Lebesgue density would integrate to0 over N,contradicting probability1",
      "relation": "implies",
      "rule_id": "contcdf_Cantor_continuous_atomless_singular_NO_density",
      "conditions": [],
      "explanation": {
        "zh-Hant": "CDF的jump等於singleton質量；右連續CDF若處處無jump便連續。滿機率零測支集證singular，且不可能由Lebesgue density積出該支集質量1；這是continuous／atomless不推出density的特定反例。",
        "en": "CDF jumps equal singleton masses;an everywhere atomless law has a continuous CDF.The full-probability null support proves singularity and rules out a Lebesgue density,since its integral over that support would be0.This is a particular counterexample to continuity/atomlessness implying 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": [
      "-13",
      "14",
      true,
      true,
      false,
      true,
      "affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density"
    ]
  },
  "conclusion": {
    "canonical_value": "(-13, 14, 是, 是, 否, 是, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)",
    "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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      "formalization": "not_formalized",
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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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  "revision": {
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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.",
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    "bilingual_semantics": true,
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    "formalization": "not_formalized",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}