Question 28159: Quadratic Reference Error Is Not a General Quadrature Error Theorem

Original question and givens

Use mode=quadratic_reference, complete model={"coefficients":[9,1,1],"nodes":[0,1,11]}. 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":[9,1,1],"nodes":[0,1,11]}. 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, [9, 1, 1]]

derived = [[9, 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, 9)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 = 9

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, 9)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 = 11

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 := mul(0, 11)

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, 11)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 = 11

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 := sum(11, 1)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 = 12

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 := mul(12, 11)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 = 132

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 := sum(132, 9)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 = 141

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

Trapezoid quadratic partition ORIGINAL width0 = 1

Trapezoid quadratic partition ORIGINAL width1 := sub(11, 1)

Trapezoid quadratic partition ORIGINAL width1 = 10

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(9, 11)

Trapezoid quadratic partition ORIGINAL endpoint sum interval0 = 20

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 := mul(1/2, 20)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 = 10

source roles = [Trapezoid quadratic partition, 0, 0, 1, 9, 11]

derived = [1, 1/2, 20, 10]

Trapezoid quadratic partition ORIGINAL width over2 interval1 := div(10, 2)

Trapezoid quadratic partition ORIGINAL width over2 interval1 = 5

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 := sum(11, 141)

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 = 152

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 := mul(5, 152)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 = 760

source roles = [Trapezoid quadratic partition, 1, 1, 11, 11, 141]

derived = [10, 5, 152, 760]

Trapezoid quadratic partition ORIGINAL trapezoid running total0 := sum(0, 10)

Trapezoid quadratic partition ORIGINAL trapezoid running total0 = 10

Trapezoid quadratic partition ORIGINAL trapezoid running total1 := sum(10, 760)

Trapezoid quadratic partition ORIGINAL trapezoid running total1 = 770

source roles = [Trapezoid quadratic partition, [[0, 1], [1, 1], [11, 1]], [[9, 1], [11, 1], [141, 1]]]

derived = [[0, 1, 11], [9, 11, 141], [1, 10], [10, 760], 770]

Trapezoid quadratic reference ORIGINAL coefficient over1 := div(9, 1)

Trapezoid quadratic reference ORIGINAL coefficient over1 = 9

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 := mul(1, 11)

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 = 11

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(11, 0)

Trapezoid quadratic reference ORIGINAL power difference0 = 11

Trapezoid quadratic reference ORIGINAL integrated term0 := mul(9, 11)

Trapezoid quadratic reference ORIGINAL integrated term0 = 99

Trapezoid quadratic reference ORIGINAL integral running sum0 := sum(0, 99)

Trapezoid quadratic reference ORIGINAL integral running sum0 = 99

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, 11)

Trapezoid quadratic reference right term1 ORIGINAL power multiply1 = 11

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 := mul(11, 11)

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 = 121

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(121, 0)

Trapezoid quadratic reference ORIGINAL power difference1 = 121

Trapezoid quadratic reference ORIGINAL integrated term1 := mul(1/2, 121)

Trapezoid quadratic reference ORIGINAL integrated term1 = 121/2

Trapezoid quadratic reference ORIGINAL integral running sum1 := sum(99, 121/2)

Trapezoid quadratic reference ORIGINAL integral running sum1 = 319/2

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, 11)

Trapezoid quadratic reference right term2 ORIGINAL power multiply1 = 11

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 := mul(11, 11)

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 = 121

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 := mul(121, 11)

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 = 1331

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(1331, 0)

Trapezoid quadratic reference ORIGINAL power difference2 = 1331

Trapezoid quadratic reference ORIGINAL integrated term2 := mul(1/3, 1331)

Trapezoid quadratic reference ORIGINAL integrated term2 = 1331/3

Trapezoid quadratic reference ORIGINAL integral running sum2 := sum(319/2, 1331/3)

Trapezoid quadratic reference ORIGINAL integral running sum2 = 3619/6

Trapezoid ORIGINAL signed error I minus T := sub(3619/6, 770)

Trapezoid ORIGINAL signed error I minus T = -1001/6

source roles = [[9, 1, 1], [0, 1, 11]]

derived = [770, 3619/6, -1001/6, 1001/6, 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, [9, 1, 1], [0, 1, 11], 770, 3619/6, -1001/6, 1001/6, 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.