第 11050 題：有限凸組合

原題與已知條件

有限非空向量列與精確有理權重。操作=convcomb-two-point；資料={"left":[12,0],"right":[15,3],"lambda":[25,27]}。請依 [left,right,lambda,in_unit_interval,point,convex_combination] 欄序回答。

• G001 — 有限非空向量列與精確有理權重。操作=convcomb-two-point；資料={"left":[12,0],"right":[15,3],"lambda":[25,27]}。請依 [left,right,lambda,in_unit_interval,point,convex_combination] 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-finite-convex-combination

逐步推導

two-point complementary weight := sub(1, 25/27)

two-point complementary weight = 2/27

two-point:raw lambda=[25, 27];lambda=25/27;one-minus-lambda=2/27

in_closed_unit_interval=是;lambda weights left

two-point:ordered vectors=[[12, 0], [15, 3]];aligned weights=[25/27, 2/27]

dimension=2;preserve duplicates and order

two-point coordinate0 term0 := mul(25/27, 12)

two-point coordinate0 term0 = 100/9

two-point coordinate0 term1 := mul(2/27, 15)

two-point coordinate0 term1 = 10/9

two-point coordinate0 sum := sum(100/9, 10/9)

two-point coordinate0 sum = 110/9

two-point coordinate1 term0 := mul(25/27, 0)

two-point coordinate1 term0 = 0

two-point coordinate1 term1 := mul(2/27, 3)

two-point coordinate1 term1 = 2/9

two-point coordinate1 sum := sum(0, 2/9)

two-point coordinate1 sum = 2/9

逐步反向核對

• S009 · premise_and_result_audit — 以精確有理數重算 two-point coordinate1 sum，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 two-point coordinate1 term1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 two-point coordinate1 term0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 two-point coordinate0 sum，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 two-point coordinate0 term1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 two-point coordinate0 term0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 以精確有理數重算 two-point complementary weight，不以小數近似替代。

結論

([12, 0], [15, 3], 25/27, 是, [110/9, 2/9], 是)

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。