第 34025 題：數值近似誤差

原題與已知條件

exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":25,"slope_a":1,"slope_b":6}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":25,"slope_a":1,"slope_b":6}；請依 exact_ordered_tuple_declared_by_interface 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-numerical-approximation-error

逐步推導

source roles = [qf-mod50-numerr-conditioning-witness, [domain, operation, residual, slope_a, slope_b]]

derived = [numerr-conditioning-witness, Exact_numerical_error_density]

source roles = [conditioning ORIGINAL residual, 25, 1]

derived = [[25, 1], 25]

conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator := div(25, 1)

conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator = 25

source roles = [conditioning ORIGINAL slope A, 1, 1]

derived = [[1, 1], 1]

conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator := div(1, 1)

conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator = 1

source roles = [conditioning ORIGINAL slope B, 6, 1]

derived = [[6, 1], 6]

conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator := div(6, 1)

conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator = 6

source roles = [A, 1]

derived = [1, 是]

source roles = [conditioning error A SOURCE naive division, 25, 1]

derived = [[25, 1], 25]

conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator := div(25, 1)

conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator = 25

conditioning error A ORIGINAL residual divided by absolute slope := div(25, 1)

conditioning error A ORIGINAL residual divided by absolute slope = 25

source roles = [B, 6]

derived = [6, 是]

source roles = [conditioning error B SOURCE naive division, 25, 6]

derived = [[25, 6], 25/6]

conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator := div(25, 6)

conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator = 25/6

conditioning error B ORIGINAL residual divided by absolute slope := div(25, 6)

conditioning error B ORIGINAL residual divided by absolute slope = 25/6

source roles = [25, 1, 6]

derived = [25, 25/6, 是]

逐步反向核對

• S016 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S015 · premise_and_result_audit — 以精確有理數重算 conditioning error B ORIGINAL residual divided by absolute slope，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator，不以小數近似替代。
• S013 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S012 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S011 · premise_and_result_audit — 以精確有理數重算 conditioning error A ORIGINAL residual divided by absolute slope，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator，不以小數近似替代。
• S009 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S008 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S007 · premise_and_result_audit — 以精確有理數重算 conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator，不以小數近似替代。
• S006 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S005 · premise_and_result_audit — 以精確有理數重算 conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S003 · premise_and_result_audit — 以精確有理數重算 conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。
• S001 · premise_and_result_audit — 以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。

結論

(25, 1, 25, 6, 25/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)

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。