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Question 28175: Quadratic Reference Error Is Not a General Quadrature Error Theorem

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

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 dependencies

Rule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

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 dependencies

Rule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

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 dependencies

Rule: Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

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 dependencies

Rule: 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