# Question 28175: Quadratic Reference Error Is Not a General Quadrature Error Theorem [Return to catalogue page 141](https://ocme.evemisslab.com/en/questions/page/141/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/#q-20261015-pulse-20261119-r02-0100) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0055-28175 · q-20261015-pulse-20261119-r02-0100 ## Original question and givens Use mode=quadratic_reference, complete model={"coefficients":[25,1,1],"nodes":[0,1,27]}. 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":[25,1,1],"nodes":[0,1,27]}. 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](https://ocme.evemisslab.com/en/mko/mko-trapezoidal-quadrature/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-trap-error-boundary, trap-error-boundary] After derived = [[mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope]] Conditions: none List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 := mul(1, 1) After Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient1 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 := mul(2, 1) After Trapezoid quadratic reference polynomial ORIGINAL derivative coefficient2 = 2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Trapezoid quadratic reference polynomial, [25, 1, 1]] After derived = [[25, 1, 1], [1, 2]] Conditions: none Require raw length2..6 before trimming trailing zeros,and remain nonconstant.Differentiate every monomial to i*c_i*x^(i-1).Source derivative coefficients must pass their1e8 profile even if a root later stops or steps=0. Step identifiers and dependenciesRule: Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients · Uses: G001, S003 S005 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 := mul(0, 0) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree2 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 := sum(0, 1) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient2 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 := mul(1, 0) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree1 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 := sum(0, 1) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient1 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 := mul(1, 0) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner multiply degree0 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 := sum(0, 25) After Trapezoid quadratic ORIGINAL node0 ORIGINAL Horner add coefficient0 = 25 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 := mul(0, 1) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree2 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 := sum(0, 1) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient2 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 := mul(1, 1) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree1 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 := sum(1, 1) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient1 = 2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 := mul(2, 1) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner multiply degree0 = 2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 := sum(2, 25) After Trapezoid quadratic ORIGINAL node1 ORIGINAL Horner add coefficient0 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 := mul(0, 27) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree2 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 := sum(0, 1) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient2 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 := mul(1, 27) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree1 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 := sum(27, 1) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient1 = 28 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 := mul(28, 27) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner multiply degree0 = 756 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 := sum(756, 25) After Trapezoid quadratic ORIGINAL node2 ORIGINAL Horner add coefficient0 = 781 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Trapezoid quadratic partition ORIGINAL width0 := sub(1, 0) After Trapezoid quadratic partition ORIGINAL width0 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Trapezoid quadratic partition ORIGINAL width1 := sub(27, 1) After Trapezoid quadratic partition ORIGINAL width1 = 26 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Trapezoid quadratic partition ORIGINAL width over2 interval0 := div(1, 2) After Trapezoid quadratic partition ORIGINAL width over2 interval0 = 1/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Trapezoid quadratic partition ORIGINAL endpoint sum interval0 := sum(25, 27) After Trapezoid quadratic partition ORIGINAL endpoint sum interval0 = 52 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 := mul(1/2, 52) After Trapezoid quadratic partition ORIGINAL individual trapezoid interval0 = 26 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore source roles = [Trapezoid quadratic partition, 0, 0, 1, 25, 27] After derived = [1, 1/2, 52, 26] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S027 S029 · calculationBefore Trapezoid quadratic partition ORIGINAL width over2 interval1 := div(26, 2) After Trapezoid quadratic partition ORIGINAL width over2 interval1 = 13 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Trapezoid quadratic partition ORIGINAL endpoint sum interval1 := sum(27, 781) After Trapezoid quadratic partition ORIGINAL endpoint sum interval1 = 808 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 := mul(13, 808) After Trapezoid quadratic partition ORIGINAL individual trapezoid interval1 = 10504 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore source roles = [Trapezoid quadratic partition, 1, 1, 27, 27, 781] After derived = [26, 13, 808, 10504] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S031 S033 · calculationBefore Trapezoid quadratic partition ORIGINAL trapezoid running total0 := sum(0, 26) After Trapezoid quadratic partition ORIGINAL trapezoid running total0 = 26 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Trapezoid quadratic partition ORIGINAL trapezoid running total1 := sum(26, 10504) After Trapezoid quadratic partition ORIGINAL trapezoid running total1 = 10530 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore source roles = [Trapezoid quadratic partition, [[0, 1], [1, 1], [27, 1]], [[25, 1], [27, 1], [781, 1]]] After derived = [[0, 1, 27], [25, 27, 781], [1, 26], [26, 10504], 10530] Conditions: none A complete partition is STRICTLY increasing in original order,with n+1 aligned nodes/values for n intervals;the24-node cap is not the vector8 cap.The table determines this sum,not a unique integral without a full integrand. Step identifiers and dependenciesRule: Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand · Uses: G001, S034 S036 · calculationBefore Trapezoid quadratic reference ORIGINAL coefficient over1 := div(25, 1) After Trapezoid quadratic reference ORIGINAL coefficient over1 = 25 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Trapezoid quadratic reference right term0 ORIGINAL power multiply1 := mul(1, 27) After Trapezoid quadratic reference right term0 ORIGINAL power multiply1 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Trapezoid quadratic reference left term0 ORIGINAL power multiply1 := mul(1, 0) After Trapezoid quadratic reference left term0 ORIGINAL power multiply1 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Trapezoid quadratic reference ORIGINAL power difference0 := sub(27, 0) After Trapezoid quadratic reference ORIGINAL power difference0 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Trapezoid quadratic reference ORIGINAL integrated term0 := mul(25, 27) After Trapezoid quadratic reference ORIGINAL integrated term0 = 675 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Trapezoid quadratic reference ORIGINAL integral running sum0 := sum(0, 675) After Trapezoid quadratic reference ORIGINAL integral running sum0 = 675 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Trapezoid quadratic reference ORIGINAL coefficient over2 := div(1, 2) After Trapezoid quadratic reference ORIGINAL coefficient over2 = 1/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Trapezoid quadratic reference right term1 ORIGINAL power multiply1 := mul(1, 27) After Trapezoid quadratic reference right term1 ORIGINAL power multiply1 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Trapezoid quadratic reference right term1 ORIGINAL power multiply2 := mul(27, 27) After Trapezoid quadratic reference right term1 ORIGINAL power multiply2 = 729 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Trapezoid quadratic reference left term1 ORIGINAL power multiply1 := mul(1, 0) After Trapezoid quadratic reference left term1 ORIGINAL power multiply1 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Trapezoid quadratic reference left term1 ORIGINAL power multiply2 := mul(0, 0) After Trapezoid quadratic reference left term1 ORIGINAL power multiply2 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Trapezoid quadratic reference ORIGINAL power difference1 := sub(729, 0) After Trapezoid quadratic reference ORIGINAL power difference1 = 729 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Trapezoid quadratic reference ORIGINAL integrated term1 := mul(1/2, 729) After Trapezoid quadratic reference ORIGINAL integrated term1 = 729/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Trapezoid quadratic reference ORIGINAL integral running sum1 := sum(675, 729/2) After Trapezoid quadratic reference ORIGINAL integral running sum1 = 2079/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Trapezoid quadratic reference ORIGINAL coefficient over3 := div(1, 3) After Trapezoid quadratic reference ORIGINAL coefficient over3 = 1/3 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Trapezoid quadratic reference right term2 ORIGINAL power multiply1 := mul(1, 27) After Trapezoid quadratic reference right term2 ORIGINAL power multiply1 = 27 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Trapezoid quadratic reference right term2 ORIGINAL power multiply2 := mul(27, 27) After Trapezoid quadratic reference right term2 ORIGINAL power multiply2 = 729 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore Trapezoid quadratic reference right term2 ORIGINAL power multiply3 := mul(729, 27) After Trapezoid quadratic reference right term2 ORIGINAL power multiply3 = 19683 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Trapezoid quadratic reference left term2 ORIGINAL power multiply1 := mul(1, 0) After Trapezoid quadratic reference left term2 ORIGINAL power multiply1 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore Trapezoid quadratic reference left term2 ORIGINAL power multiply2 := mul(0, 0) After Trapezoid quadratic reference left term2 ORIGINAL power multiply2 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Trapezoid quadratic reference left term2 ORIGINAL power multiply3 := mul(0, 0) After Trapezoid quadratic reference left term2 ORIGINAL power multiply3 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Trapezoid quadratic reference ORIGINAL power difference2 := sub(19683, 0) After Trapezoid quadratic reference ORIGINAL power difference2 = 19683 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Trapezoid quadratic reference ORIGINAL integrated term2 := mul(1/3, 19683) After Trapezoid quadratic reference ORIGINAL integrated term2 = 6561 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Trapezoid quadratic reference ORIGINAL integral running sum2 := sum(2079/2, 6561) After Trapezoid quadratic reference ORIGINAL integral running sum2 = 15201/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Trapezoid ORIGINAL signed error I minus T := sub(15201/2, 10530) After Trapezoid ORIGINAL signed error I minus T = -5859/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore source roles = [[25, 1, 1], [0, 1, 27]] After derived = [10530, 15201/2, -5859/2, 5859/2, No] Conditions: none The supplied WHOLE quadratic permits an exact reference integral.Signed error is I-T with absolute error separate;this neither infers a unique integrand from samples nor supplies a general second-derivative/error-order theorem. Step identifiers and dependenciesRule: Trapezoid_ACTUAL_full_quadratic_reference_IminusT_NOT_general_error_theorem · Uses: G001, S060 ## 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, [25, 1, 1], [0, 1, 27], 10530, 15201/2, -5859/2, 5859/2, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 040718e3f916e9130794c08f7b4cd4ca8985d5c6d3595582a98236d1801f8ca0 - candidate record SHA-256: 3ace5564e99d90846845754c084eae8bac4576d12bab843c5c33736a84f38202 - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r02-0100.json)