Question 28155: Quadratic Reference Error Is Not a General Quadrature Error Theorem

Original question and givens

Use mode=quadratic_reference, complete model={"coefficients":[5,1,1],"nodes":[0,1,7]}. 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":[5,1,1],"nodes":[0,1,7]}. 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, [5, 1, 1]]

derived = [[5, 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, 5)

Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 = 5

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, 5)

Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 = 7

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 := mul(0, 7)

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, 7)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 = 7

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 := sum(7, 1)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 = 8

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 := mul(8, 7)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 = 56

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 := sum(56, 5)

Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 = 61

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

Trapezoid quadratic partition ORIGINAL width0 = 1

Trapezoid quadratic partition ORIGINAL width1 := sub(7, 1)

Trapezoid quadratic partition ORIGINAL width1 = 6

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(5, 7)

Trapezoid quadratic partition ORIGINAL endpoint sum interval0 = 12

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 := mul(1/2, 12)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 = 6

source roles = [Trapezoid quadratic partition, 0, 0, 1, 5, 7]

derived = [1, 1/2, 12, 6]

Trapezoid quadratic partition ORIGINAL width over2 interval1 := div(6, 2)

Trapezoid quadratic partition ORIGINAL width over2 interval1 = 3

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 := sum(7, 61)

Trapezoid quadratic partition ORIGINAL endpoint sum interval1 = 68

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 := mul(3, 68)

Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 = 204

source roles = [Trapezoid quadratic partition, 1, 1, 7, 7, 61]

derived = [6, 3, 68, 204]

Trapezoid quadratic partition ORIGINAL trapezoid running total0 := sum(0, 6)

Trapezoid quadratic partition ORIGINAL trapezoid running total0 = 6

Trapezoid quadratic partition ORIGINAL trapezoid running total1 := sum(6, 204)

Trapezoid quadratic partition ORIGINAL trapezoid running total1 = 210

source roles = [Trapezoid quadratic partition, [[0, 1], [1, 1], [7, 1]], [[5, 1], [7, 1], [61, 1]]]

derived = [[0, 1, 7], [5, 7, 61], [1, 6], [6, 204], 210]

Trapezoid quadratic reference ORIGINAL coefficient over1 := div(5, 1)

Trapezoid quadratic reference ORIGINAL coefficient over1 = 5

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 := mul(1, 7)

Trapezoid quadratic reference right term0 ORIGINAL power multiply1 = 7

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(7, 0)

Trapezoid quadratic reference ORIGINAL power difference0 = 7

Trapezoid quadratic reference ORIGINAL integrated term0 := mul(5, 7)

Trapezoid quadratic reference ORIGINAL integrated term0 = 35

Trapezoid quadratic reference ORIGINAL integral running sum0 := sum(0, 35)

Trapezoid quadratic reference ORIGINAL integral running sum0 = 35

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, 7)

Trapezoid quadratic reference right term1 ORIGINAL power multiply1 = 7

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 := mul(7, 7)

Trapezoid quadratic reference right term1 ORIGINAL power multiply2 = 49

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(49, 0)

Trapezoid quadratic reference ORIGINAL power difference1 = 49

Trapezoid quadratic reference ORIGINAL integrated term1 := mul(1/2, 49)

Trapezoid quadratic reference ORIGINAL integrated term1 = 49/2

Trapezoid quadratic reference ORIGINAL integral running sum1 := sum(35, 49/2)

Trapezoid quadratic reference ORIGINAL integral running sum1 = 119/2

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, 7)

Trapezoid quadratic reference right term2 ORIGINAL power multiply1 = 7

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 := mul(7, 7)

Trapezoid quadratic reference right term2 ORIGINAL power multiply2 = 49

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 := mul(49, 7)

Trapezoid quadratic reference right term2 ORIGINAL power multiply3 = 343

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(343, 0)

Trapezoid quadratic reference ORIGINAL power difference2 = 343

Trapezoid quadratic reference ORIGINAL integrated term2 := mul(1/3, 343)

Trapezoid quadratic reference ORIGINAL integrated term2 = 343/3

Trapezoid quadratic reference ORIGINAL integral running sum2 := sum(119/2, 343/3)

Trapezoid quadratic reference ORIGINAL integral running sum2 = 1043/6

Trapezoid ORIGINAL signed error I minus T := sub(1043/6, 210)

Trapezoid ORIGINAL signed error I minus T = -217/6

source roles = [[5, 1, 1], [0, 1, 7]]

derived = [210, 1043/6, -217/6, 217/6, 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, [5, 1, 1], [0, 1, 7], 210, 1043/6, -217/6, 217/6, 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.