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Question 28153: 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

R0055-28153 · q-20261015-pulse-20261119-r02-0078

Original question and givens

Use mode=sample_table_scope, complete model={"nodes":[0,1,5],"values":[2,3,5]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).

  • G001 — Use mode=sample_table_scope, complete model={"nodes":[0,1,5],"values":[2,3,5]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).

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, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary]]

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 boundary ORIGINAL ORIGINAL width0 := sub(1, 0)

After

Trapezoid boundary ORIGINAL 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, S001

S003 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL width1 := sub(5, 1)

After

Trapezoid boundary ORIGINAL ORIGINAL width1 = 4

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

After

Trapezoid boundary ORIGINAL 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, S003

S005 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 := sum(2, 3)

After

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 = 5

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 boundary ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 5)

After

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 = 5/2

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, S005

S007 · implies

Before

source roles = [Trapezoid boundary ORIGINAL, 0, 0, 1, 2, 3]

After

derived = [1, 1/2, 5, 5/2]

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, S006

S008 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 := div(4, 2)

After

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 = 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, S007

S009 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 := sum(3, 5)

After

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 = 8

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 boundary ORIGINAL ORIGINAL individual trapezoid interval1 := mul(2, 8)

After

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 = 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, S009

S011 · implies

Before

source roles = [Trapezoid boundary ORIGINAL, 1, 1, 5, 3, 5]

After

derived = [4, 2, 8, 16]

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, S010

S012 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 5/2)

After

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 = 5/2

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 · calculation

Before

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 := sum(5/2, 16)

After

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 = 37/2

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, S012

S014 · implies

Before

source roles = [Trapezoid boundary ORIGINAL, [0, 1, 5], [2, 3, 5]]

After

derived = [[0, 1, 5], [2, 3, 5], [1, 4], [5/2, 16], 37/2]

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, S013

S015 · implies

Before

source roles = [[0, 1, 5], [2, 3, 5]]

After

derived = [37/2, Yes, No]

Conditions: none

Finite endpoint values determine the quadrature sum.Adding a nonzero function vanishing at every node leaves samples unchanged,so they do not ensure a unique integrand or integral.

Step identifiers and dependencies

Rule: Trapezoid_sample_table_sum_ONLY_not_unique_full_integrand · Uses: G001, S014

Reverse checks

  • S015 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S014 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S013 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S010 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S006 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Trapezoid boundary ORIGINAL ORIGINAL width0 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

(sample_table_scope, [0, 1, 5], [2, 3, 5], 37/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)

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: 7331db3a4dee2f653bdb3b52f453473c7c8c779abbb42b2b3b3e570b66a7c06f
  • candidate record SHA-256: 9028afb5618dae50393055b2b1213e7faf457f3e279e35d6ae9cd8d273cb8a5a
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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