Question 28121: An Affine Integrand Is Integrated Exactly on Any Qualified Partition

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0055-28121 · q-20261015-pulse-20261119-r02-0071

Original question and givens

Use f(x)=m*x+b with m=3, b=21 and strictly increasing nodes=[0, 1, 23]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).

G001 — Use f(x)=m*x+b with m=3, b=21 and strictly increasing nodes=[0, 1, 23]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, 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

S001 · implies

Before

source roles = [qf-mod50-trap-affine-exactness, trap-affine-exactness]

After

derived = [[slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, 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 affine ORIGINAL value product0 := mul(3, 0)

After

Trapezoid affine ORIGINAL value product0 = 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, S001

S003 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept0 := sum(0, 21)

After

Trapezoid affine ORIGINAL value plus intercept0 = 21

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 · calculation

Before

Trapezoid affine ORIGINAL value product1 := mul(3, 1)

After

Trapezoid affine ORIGINAL value product1 = 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, S003

S005 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept1 := sum(3, 21)

After

Trapezoid affine ORIGINAL value plus intercept1 = 24

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 affine ORIGINAL value product2 := mul(3, 23)

After

Trapezoid affine ORIGINAL value product2 = 69

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 affine ORIGINAL value plus intercept2 := sum(69, 21)

After

Trapezoid affine ORIGINAL value plus intercept2 = 90

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 affine partition ORIGINAL width0 := sub(1, 0)

After

Trapezoid affine 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, S007

S009 · calculation

Before

Trapezoid affine partition ORIGINAL width1 := sub(23, 1)

After

Trapezoid affine partition ORIGINAL width1 = 22

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

After

Trapezoid affine 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, S009

S011 · calculation

Before

Trapezoid affine partition ORIGINAL endpoint sum interval0 := sum(21, 24)

After

Trapezoid affine partition ORIGINAL endpoint sum interval0 = 45

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 affine partition ORIGINAL individual trapezoid interval0 := mul(1/2, 45)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval0 = 45/2

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, S011

S013 · implies

Before

source roles = [Trapezoid affine partition, 0, 0, 1, 21, 24]

After

derived = [1, 1/2, 45, 45/2]

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, S012

S014 · calculation

Before

Trapezoid affine partition ORIGINAL width over2 interval1 := div(22, 2)

After

Trapezoid affine partition ORIGINAL width over2 interval1 = 11

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 affine partition ORIGINAL endpoint sum interval1 := sum(24, 90)

After

Trapezoid affine partition ORIGINAL endpoint sum interval1 = 114

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 affine partition ORIGINAL individual trapezoid interval1 := mul(11, 114)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval1 = 1254

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 · implies

Before

source roles = [Trapezoid affine partition, 1, 1, 23, 24, 90]

After

derived = [22, 11, 114, 1254]

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, S016

S018 · calculation

Before

Trapezoid affine partition ORIGINAL trapezoid running total0 := sum(0, 45/2)

After

Trapezoid affine partition ORIGINAL trapezoid running total0 = 45/2

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, S017

S019 · calculation

Before

Trapezoid affine partition ORIGINAL trapezoid running total1 := sum(45/2, 1254)

After

Trapezoid affine partition ORIGINAL trapezoid running total1 = 2553/2

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, S018

S020 · implies

Before

source roles = [Trapezoid affine partition, [[0, 1], [1, 1], [23, 1]], [[21, 1], [24, 1], [90, 1]]]

After

derived = [[0, 1, 23], [21, 24, 90], [1, 22], [45/2, 1254], 2553/2]

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, S019

S021 · calculation

Before

Trapezoid affine ORIGINAL slope over2 := div(3, 2)

After

Trapezoid affine ORIGINAL slope over2 = 3/2

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, S020

S022 · calculation

Before

Trapezoid affine ORIGINAL right square := mul(23, 23)

After

Trapezoid affine ORIGINAL right square = 529

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 affine ORIGINAL left square := mul(0, 0)

After

Trapezoid affine ORIGINAL left square = 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, S022

S024 · calculation

Before

Trapezoid affine ORIGINAL square difference := sub(529, 0)

After

Trapezoid affine ORIGINAL square difference = 529

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 affine ORIGINAL quadratic integral part := mul(3/2, 529)

After

Trapezoid affine ORIGINAL quadratic integral part = 1587/2

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 affine ORIGINAL whole width := sub(23, 0)

After

Trapezoid affine ORIGINAL whole width = 23

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 affine ORIGINAL intercept integral part := mul(21, 23)

After

Trapezoid affine ORIGINAL intercept integral part = 483

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 · calculation

Before

Trapezoid affine ORIGINAL exact integral := sum(1587/2, 483)

After

Trapezoid affine ORIGINAL exact integral = 2553/2

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, S027

S029 · implies

Before

source roles = [3, 21, [0, 1, 23], [21, 24, 90]]

After

derived = [2553/2, 2553/2, Yes]

Conditions: none

The WHOLE affine law makes each trapezoid equal to the antiderivative difference over the same interval.Internal endpoints telescope,hence exactness on any qualified partition;not for arbitrary functions.

Step identifiers and dependencies

Rule: Trapezoid_WHOLE_affine_exactness_each_interval_antiderivative_telescopes · Uses: G001, S028

Reverse checks

S029 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S028 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL exact integral with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL intercept integral part with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL whole width with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL quadratic integral part with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL square difference with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL left square with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL right square with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL slope over2 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S019 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S016 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S012 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept2 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product2 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product1 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept0 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product0 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

(3, 21, [0, 1, 23], [21, 24, 90], 2553/2, 2553/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)

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: 9af17d18ba104b1b6345a9a28d05eae406d75283e112e4e1dfa1daeaf4756a26
candidate record SHA-256: 42e1b43c0542c22a12c7c110d1d170bc8afbad6f0dbbcdcd1de62a1fe708b72f
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question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
Lean: NOT_RUN · human review: NOT_PERFORMED

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