Question 28163: Quadratic Reference Error Is Not a General Quadrature Error Theorem

Original question and givens

Use mode=quadratic_reference, complete model={"coefficients":[13,1,1],"nodes":[0,1,15]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).

• G001 — Use mode=quadratic_reference, complete model={"coefficients":[13,1,1],"nodes":[0,1,15]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).

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, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope]]

Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 := mul(1, 1)

Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 = 1

Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 := mul(2, 1)

Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 = 2

source roles = [Trapezoid quadratic reference polynomial, [13, 1, 1]]

derived = [[13, 1, 1], [1, 2]]

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 := mul(0, 0)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 = 0

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 := sum(0, 1)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 = 1

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 := mul(1, 0)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 = 0

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 := sum(0, 1)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 = 1

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 := mul(1, 0)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 = 0

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 := sum(0, 13)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 = 13

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 := mul(0, 1)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 = 0

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 := sum(0, 1)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 = 1

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 := mul(1, 1)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 = 1

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 := sum(1, 1)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 = 2

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 := mul(2, 1)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 = 2

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 := sum(2, 13)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 = 15

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 := mul(0, 15)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 = 0

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 := sum(0, 1)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 = 1

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 := mul(1, 15)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 = 15

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 := sum(15, 1)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 = 16

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 := mul(16, 15)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 = 240

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 := sum(240, 13)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 = 253

Trapezoid quadratic partition ORIGINAL width0 := sub(1, 0)

Trapezoid quadratic partition ORIGINAL width0 = 1

Trapezoid quadratic partition ORIGINAL width1 := sub(15, 1)

Trapezoid quadratic partition ORIGINAL width1 = 14

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

Trapezoid quadratic partition ORIGINAL width over2 interval0 = 1/2

Trapezoid quadratic partition ORIGINAL endpoint sum interval0 := sum(13, 15)

Trapezoid quadratic partition ORIGINAL endpoint sum interval0 = 28

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 := mul(1/2, 28)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 = 14

source roles = [Trapezoid quadratic partition, 0, 0, 1, 13, 15]

derived = [1, 1/2, 28, 14]

Trapezoid quadratic partition ORIGINAL width over2 interval1 := div(14, 2)

Trapezoid quadratic partition ORIGINAL width over2 interval1 = 7

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 := sum(15, 253)

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 = 268

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 := mul(7, 268)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 = 1876

source roles = [Trapezoid quadratic partition, 1, 1, 15, 15, 253]

derived = [14, 7, 268, 1876]

Trapezoid quadratic partition ORIGINAL trapezoid running total0 := sum(0, 14)

Trapezoid quadratic partition ORIGINAL trapezoid running total0 = 14

Trapezoid quadratic partition ORIGINAL trapezoid running total1 := sum(14, 1876)

Trapezoid quadratic partition ORIGINAL trapezoid running total1 = 1890

source roles = [Trapezoid quadratic partition, [[0, 1], [1, 1], [15, 1]], [[13, 1], [15, 1], [253, 1]]]

derived = [[0, 1, 15], [13, 15, 253], [1, 14], [14, 1876], 1890]

Trapezoid quadratic reference ORIGINAL coefficient over1 := div(13, 1)

Trapezoid quadratic reference ORIGINAL coefficient over1 = 13

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 := mul(1, 15)

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 = 15

Trapezoid quadratic reference left term0 ORIGINAL power multiply1 := mul(1, 0)

Trapezoid quadratic reference left term0 ORIGINAL power multiply1 = 0

Trapezoid quadratic reference ORIGINAL power difference0 := sub(15, 0)

Trapezoid quadratic reference ORIGINAL power difference0 = 15

Trapezoid quadratic reference ORIGINAL integrated term0 := mul(13, 15)

Trapezoid quadratic reference ORIGINAL integrated term0 = 195

Trapezoid quadratic reference ORIGINAL integral running sum0 := sum(0, 195)

Trapezoid quadratic reference ORIGINAL integral running sum0 = 195

Trapezoid quadratic reference ORIGINAL coefficient over2 := div(1, 2)

Trapezoid quadratic reference ORIGINAL coefficient over2 = 1/2

Trapezoid quadratic reference right term1 ORIGINAL power multiply1 := mul(1, 15)

Trapezoid quadratic reference right term1 ORIGINAL power multiply1 = 15

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 := mul(15, 15)

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 = 225

Trapezoid quadratic reference left term1 ORIGINAL power multiply1 := mul(1, 0)

Trapezoid quadratic reference left term1 ORIGINAL power multiply1 = 0

Trapezoid quadratic reference left term1 ORIGINAL power multiply2 := mul(0, 0)

Trapezoid quadratic reference left term1 ORIGINAL power multiply2 = 0

Trapezoid quadratic reference ORIGINAL power difference1 := sub(225, 0)

Trapezoid quadratic reference ORIGINAL power difference1 = 225

Trapezoid quadratic reference ORIGINAL integrated term1 := mul(1/2, 225)

Trapezoid quadratic reference ORIGINAL integrated term1 = 225/2

Trapezoid quadratic reference ORIGINAL integral running sum1 := sum(195, 225/2)

Trapezoid quadratic reference ORIGINAL integral running sum1 = 615/2

Trapezoid quadratic reference ORIGINAL coefficient over3 := div(1, 3)

Trapezoid quadratic reference ORIGINAL coefficient over3 = 1/3

Trapezoid quadratic reference right term2 ORIGINAL power multiply1 := mul(1, 15)

Trapezoid quadratic reference right term2 ORIGINAL power multiply1 = 15

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 := mul(15, 15)

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 = 225

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 := mul(225, 15)

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 = 3375

Trapezoid quadratic reference left term2 ORIGINAL power multiply1 := mul(1, 0)

Trapezoid quadratic reference left term2 ORIGINAL power multiply1 = 0

Trapezoid quadratic reference left term2 ORIGINAL power multiply2 := mul(0, 0)

Trapezoid quadratic reference left term2 ORIGINAL power multiply2 = 0

Trapezoid quadratic reference left term2 ORIGINAL power multiply3 := mul(0, 0)

Trapezoid quadratic reference left term2 ORIGINAL power multiply3 = 0

Trapezoid quadratic reference ORIGINAL power difference2 := sub(3375, 0)

Trapezoid quadratic reference ORIGINAL power difference2 = 3375

Trapezoid quadratic reference ORIGINAL integrated term2 := mul(1/3, 3375)

Trapezoid quadratic reference ORIGINAL integrated term2 = 1125

Trapezoid quadratic reference ORIGINAL integral running sum2 := sum(615/2, 1125)

Trapezoid quadratic reference ORIGINAL integral running sum2 = 2865/2

Trapezoid ORIGINAL signed error I minus T := sub(2865/2, 1890)

Trapezoid ORIGINAL signed error I minus T = -915/2

source roles = [[13, 1, 1], [0, 1, 15]]

derived = [1890, 2865/2, -915/2, 915/2, No]

Reverse checks

• S061 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S060 · premise_and_result_audit — Recompute Trapezoid ORIGINAL signed error I minus T with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integral running sum2 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integrated term2 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL power difference2 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term2 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term2 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL coefficient over3 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integral running sum1 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integrated term1 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL power difference1 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term1 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term1 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term1 ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term1 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL coefficient over2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integral running sum0 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL integrated term0 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL power difference0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Trapezoid quadratic reference left term0 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Trapezoid quadratic reference right term0 ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Trapezoid quadratic reference ORIGINAL coefficient over1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S034 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S031 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S027 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Trapezoid quadratic partition ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S003 · premise_and_result_audit — Recompute Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 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

(quadratic_reference, [13, 1, 1], [0, 1, 15], 1890, 2865/2, -915/2, 915/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)

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.