R0044-22862 · q-20261014-advance-20261111-r04-continuous-real-distribution-function-0087
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":-12,"right":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 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":-12,"right":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
操作前
source operation=contcdf-singular-no-density-boundary;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[left, right, CDF_continuous, atomless, has_density, singular, boundary]
條件：無
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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
Cantor affine support positive width := sub(13, -12)
操作後
Cantor affine support positive width = 25
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S001
操作前
ORIGINAL affine Cantor support[-12,13];positive width W=25
操作後
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條件：無
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。
規則：contcdf_Cantor_WHOLE_binary_digit_construction_not_knots · 依賴：G001, S002
操作前
ORIGINAL support width W=25;level n ternary construction uses2^n intervals of length W/3^n
操作後
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
條件：無
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。
規則：contcdf_Cantor_null_support_and_vanishing_atom_bounds · 依賴：G001, S003
操作前
ORIGINAL affine law on[-12,13] has support N with Lebesgue measure0 and probability1;all singleton probabilities0
操作後
CDF_continuous=是;atomless=是;has_density=否;singular=是;any Lebesgue density would integrate to0 over N,contradicting probability1
條件：無
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。
規則：contcdf_Cantor_continuous_atomless_singular_NO_density · 依賴：G001, S004
S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 以精確有理數重算 Cantor affine support positive width，不以小數近似替代。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。(-12, 13, 是, 是, 否, 是, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
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。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: cd9f138740b4abb18ae01417ee5cdc4eef2fa606d439a4e3e3b563243ee33d79candidate record SHA-256: 559d4d9b18602af44644b9789932f664f8eb085930d94a8a75f4a900edcc8638semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7Lean: NOT_RUN · human review: NOT_PERFORMED