第 28166 題：複合梯形法

原題與已知條件

嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={"mode":"single_width_confusion","model":{"nodes":[0,1,18],"values":[15,31,49]}}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — 嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={"mode":"single_width_confusion","model":{"nodes":[0,1,18],"values":[15,31,49]}}；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-trapezoidal-quadrature

逐步推導

source roles = [qf-mod50-trap-error-boundary, trap-error-boundary]

derived = [[mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary]]

Trapezoid boundary ORIGINAL ORIGINAL width0 := sub(1, 0)

Trapezoid boundary ORIGINAL ORIGINAL width0 = 1

Trapezoid boundary ORIGINAL ORIGINAL width1 := sub(18, 1)

Trapezoid boundary ORIGINAL ORIGINAL width1 = 17

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 := div(1, 2)

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 = 1/2

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 := sum(15, 31)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 = 46

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 46)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 = 23

source roles = [Trapezoid boundary ORIGINAL, 0, 0, 1, 15, 31]

derived = [1, 1/2, 46, 23]

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 := div(17, 2)

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 = 17/2

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 := sum(31, 49)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 = 80

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 := mul(17/2, 80)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 = 680

source roles = [Trapezoid boundary ORIGINAL, 1, 1, 18, 31, 49]

derived = [17, 17/2, 80, 680]

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 23)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 = 23

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 := sum(23, 680)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 = 703

source roles = [Trapezoid boundary ORIGINAL, [0, 1, 18], [15, 31, 49]]

derived = [[0, 1, 18], [15, 31, 49], [1, 17], [23, 680], 703]

Trapezoid wrongwidth ORIGINAL end values sum := sum(15, 49)

Trapezoid wrongwidth ORIGINAL end values sum = 64

Trapezoid wrongwidth ORIGINAL end values half := div(64, 2)

Trapezoid wrongwidth ORIGINAL end values half = 32

Trapezoid wrongwidth ORIGINAL interior sum0 := sum(0, 31)

Trapezoid wrongwidth ORIGINAL interior sum0 = 31

Trapezoid wrongwidth ORIGINAL bracket := sum(32, 31)

Trapezoid wrongwidth ORIGINAL bracket = 63

Trapezoid wrongwidth ORIGINAL FIRST width times bracket := mul(1, 63)

Trapezoid wrongwidth ORIGINAL FIRST width times bracket = 63

source roles = [[1, 17], [15, 31, 49]]

derived = [703, 63, 否]

逐步反向核對

• S020 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S019 · premise_and_result_audit — 以精確有理數重算 Trapezoid wrongwidth ORIGINAL FIRST width times bracket，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Trapezoid wrongwidth ORIGINAL bracket，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Trapezoid wrongwidth ORIGINAL interior sum0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Trapezoid wrongwidth ORIGINAL end values half，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Trapezoid wrongwidth ORIGINAL end values sum，不以小數近似替代。
• S014 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S013 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0，不以小數近似替代。
• S011 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S010 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1，不以小數近似替代。
• S007 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S006 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。

結論

(single_width_confusion, [1, 17], 703, 63, 否, nonuniform_partition_cannot_use_one_global_width)

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。