第 22819 題：連續實值分布函數

原題與已知條件

CDF本身continuous、nondecreasing、right-continuous且audit query_points嚴格遞增。操作=contcdf-monotone-endpoint-audit；資料={"knots":[-20,0,20],"values":[0,[19,20],1],"query_points":[-21,0,21]}。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-monotone-endpoint-audit；資料={"knots":[-20,0,20],"values":[0,[19,20],1],"query_points":[-21,0,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-continuous-real-distribution-function

逐步推導

source operation=contcdf-monotone-endpoint-audit;original snapshot and mathematics retained

explicit ordered source output fields=[query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate]

CDF segment0 positive knot width := sub(0, -20)

CDF segment0 positive knot width = 20

CDF segment0 nonnegative value increment := sub(19/20, 0)

CDF segment0 nonnegative value increment = 19/20

CDF segment0 exact nonnegative slope := div(19/20, 20)

CDF segment0 exact nonnegative slope = 19/400

CDF segment1 positive knot width := sub(20, 0)

CDF segment1 positive knot width = 20

CDF segment1 nonnegative value increment := sub(1, 19/20)

CDF segment1 nonnegative value increment = 1/20

CDF segment1 exact nonnegative slope := div(1/20, 20)

CDF segment1 exact nonnegative slope = 1/400

ORIGINAL strictly increasing knots=[-20, 0, 20];aligned exact nondecreasing values=[0, 19/20, 1];ALL [width,height,slope] rows=[[20, 19/20, 19/400], [20, 1/20, 1/400]]

declare linear interpolation on EVERY segment,constant0 left and1 right;shared knot values make joins continuous;all slopes>=0 allow flats;continuous implies right-continuous and endpoints give limits0/1

CDF ORIGINAL ordered query0:ORIGINAL exact query x=-21;knots=[-20, 0, 20];values=[0, 19/20, 1]

case=left constant tail including first knot;source interval index=null;F(x)=0;this is CDF value,NOT singleton mass

CDF ORIGINAL ordered query1 ORIGINAL query minus left knot := sub(0, -20)

CDF ORIGINAL ordered query1 ORIGINAL query minus left knot = 20

CDF ORIGINAL ordered query1 exact interval fraction := div(20, 20)

CDF ORIGINAL ordered query1 exact interval fraction = 1

CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment := mul(1, 19/20)

CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment = 19/20

CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value := sum(0, 19/20)

CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value = 19/20

CDF ORIGINAL ordered query1:ORIGINAL exact query x=0;knots=[-20, 0, 20];values=[0, 19/20, 1]

case=FIRST source interval containing query,including shared knot;source interval index=0;F(x)=19/20;this is CDF value,NOT singleton mass

CDF ORIGINAL ordered query2:ORIGINAL exact query x=21;knots=[-20, 0, 20];values=[0, 19/20, 1]

case=right constant tail including last knot;source interval index=null;F(x)=1;this is CDF value,NOT singleton mass

CDF ORIGINAL left-tail check:ORIGINAL exact query x=-21;knots=[-20, 0, 20];values=[0, 19/20, 1]

case=left constant tail including first knot;source interval index=null;F(x)=0;this is CDF value,NOT singleton mass

CDF ORIGINAL right-tail check:ORIGINAL exact query x=21;knots=[-20, 0, 20];values=[0, 19/20, 1]

case=right constant tail including last knot;source interval index=null;F(x)=1;this is CDF value,NOT singleton mass

ORIGINAL strictly increasing queries=[-21, 0, 21];exact values=[0, 19/20, 1]

NONSTRICT sample value monotonicity=是;left tail=0,right tail=1;global monotonicity already proved from the WHOLE law

逐步反向核對

• S018 · 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。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 CDF ORIGINAL ordered query1 exact interval fraction，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 CDF ORIGINAL ordered query1 ORIGINAL query minus left knot，不以小數近似替代。
• S009 · 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。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 CDF segment1 exact nonnegative slope，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 CDF segment1 nonnegative value increment，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 CDF segment1 positive knot width，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 CDF segment0 exact nonnegative slope，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 CDF segment0 nonnegative value increment，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 CDF segment0 positive knot 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。

結論

([[-21, 0], [0, 19/20], [21, 1]], 是, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)

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。

本題覆核與來源綁定

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。