Question 28116: 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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R0055-28116 · q-20261015-pulse-20261119-r02-0066

Original question and givens

Use f(x)=m*x+b with m=-2, b=16 and strictly increasing nodes=[0, 1, 18]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).

G001 — Use f(x)=m*x+b with m=-2, b=16 and strictly increasing nodes=[0, 1, 18]. 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(-2, 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, 16)

After

Trapezoid affine ORIGINAL value plus intercept0 = 16

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(-2, 1)

After

Trapezoid affine ORIGINAL value product1 = -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, S003

S005 · calculation

Before

Trapezoid affine ORIGINAL value plus intercept1 := sum(-2, 16)

After

Trapezoid affine ORIGINAL value plus intercept1 = 14

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(-2, 18)

After

Trapezoid affine ORIGINAL value product2 = -36

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(-36, 16)

After

Trapezoid affine ORIGINAL value plus intercept2 = -20

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(18, 1)

After

Trapezoid affine partition ORIGINAL width1 = 17

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(16, 14)

After

Trapezoid affine partition ORIGINAL endpoint sum interval0 = 30

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, 30)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval0 = 15

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, 16, 14]

After

derived = [1, 1/2, 30, 15]

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(17, 2)

After

Trapezoid affine partition ORIGINAL width over2 interval1 = 17/2

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 affine partition ORIGINAL endpoint sum interval1 := sum(14, -20)

After

Trapezoid affine partition ORIGINAL endpoint sum interval1 = -6

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(17/2, -6)

After

Trapezoid affine partition ORIGINAL individual trapezoid interval1 = -51

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, 18, 14, -20]

After

derived = [17, 17/2, -6, -51]

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, 15)

After

Trapezoid affine partition ORIGINAL trapezoid running total0 = 15

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(15, -51)

After

Trapezoid affine partition ORIGINAL trapezoid running total1 = -36

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], [18, 1]], [[16, 1], [14, 1], [-20, 1]]]

After

derived = [[0, 1, 18], [16, 14, -20], [1, 17], [15, -51], -36]

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(-2, 2)

After

Trapezoid affine ORIGINAL slope over2 = -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, S020

S022 · calculation

Before

Trapezoid affine ORIGINAL right square := mul(18, 18)

After

Trapezoid affine ORIGINAL right square = 324

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(324, 0)

After

Trapezoid affine ORIGINAL square difference = 324

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(-1, 324)

After

Trapezoid affine ORIGINAL quadratic integral part = -324

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(18, 0)

After

Trapezoid affine ORIGINAL whole width = 18

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(16, 18)

After

Trapezoid affine ORIGINAL intercept integral part = 288

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(-324, 288)

After

Trapezoid affine ORIGINAL exact integral = -36

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 = [-2, 16, [0, 1, 18], [16, 14, -20]]

After

derived = [-36, -36, 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

(-2, 16, [0, 1, 18], [16, 14, -20], -36, -36, 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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 584db5b5964d726d0f32656a37452d30ca9a42f49c044a076139cfb4f625d909
candidate record SHA-256: dc7f0b39191d81999a3504b3ba18aeb6e0b1d8c1fcc187079d503eee64f36c50
semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
Lean: NOT_RUN · human review: NOT_PERFORMED

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