Question 28154: Quadratic Reference Error Is Not a General Quadrature Error Theorem

Original question and givens

Use mode=single_width_confusion, complete model={"nodes":[0,1,6],"values":[3,7,13]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).

• G001 — Use mode=single_width_confusion, complete model={"nodes":[0,1,6],"values":[3,7,13]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-trapezoidal-quadrature

Step-by-step derivation

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(6, 1)

Trapezoid boundary ORIGINAL ORIGINAL width1 = 5

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(3, 7)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 = 10

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 10)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 = 5

source roles = [Trapezoid boundary ORIGINAL, 0, 0, 1, 3, 7]

derived = [1, 1/2, 10, 5]

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 := div(5, 2)

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 = 5/2

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 := sum(7, 13)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 = 20

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 := mul(5/2, 20)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 = 50

source roles = [Trapezoid boundary ORIGINAL, 1, 1, 6, 7, 13]

derived = [5, 5/2, 20, 50]

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 5)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 = 5

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 := sum(5, 50)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 = 55

source roles = [Trapezoid boundary ORIGINAL, [0, 1, 6], [3, 7, 13]]

derived = [[0, 1, 6], [3, 7, 13], [1, 5], [5, 50], 55]

Trapezoid wrongwidth ORIGINAL end values sum := sum(3, 13)

Trapezoid wrongwidth ORIGINAL end values sum = 16

Trapezoid wrongwidth ORIGINAL end values half := div(16, 2)

Trapezoid wrongwidth ORIGINAL end values half = 8

Trapezoid wrongwidth ORIGINAL interior sum0 := sum(0, 7)

Trapezoid wrongwidth ORIGINAL interior sum0 = 7

Trapezoid wrongwidth ORIGINAL bracket := sum(8, 7)

Trapezoid wrongwidth ORIGINAL bracket = 15

Trapezoid wrongwidth ORIGINAL FIRST width times bracket := mul(1, 15)

Trapezoid wrongwidth ORIGINAL FIRST width times bracket = 15

source roles = [[1, 5], [3, 7, 13]]

derived = [55, 15, No]

Reverse checks

• S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S019 · premise_and_result_audit — Recompute Trapezoid wrongwidth ORIGINAL FIRST width times bracket with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Trapezoid wrongwidth ORIGINAL bracket with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Trapezoid wrongwidth ORIGINAL interior sum0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Trapezoid wrongwidth ORIGINAL end values half with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Trapezoid wrongwidth ORIGINAL end values sum with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S013 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S010 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S006 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

(single_width_confusion, [1, 5], 55, 15, No, nonuniform_partition_cannot_use_one_global_width)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.