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Question 28181: n Subintervals Require n+1 Aligned Strictly Increasing Nodes

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

R0055-28181 · q-20261015-pulse-20261119-r02-0006

Original question and givens

Use ordered nodes=[0, 1, 8] and aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).

  • G001 — Use ordered nodes=[0, 1, 8] and aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).

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-partition-qualification, trap-partition-qualification]

After

derived = [[nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention]]

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 ORIGINAL ORIGINAL width0 := sub(1, 0)

After

Trapezoid 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 ORIGINAL ORIGINAL width1 := sub(8, 1)

After

Trapezoid ORIGINAL ORIGINAL width1 = 7

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

After

Trapezoid 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 ORIGINAL ORIGINAL endpoint sum interval0 := sum(5, 6)

After

Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 = 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, S004

S006 · calculation

Before

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 11)

After

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 = 11/2

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 ORIGINAL, 0, 0, 1, 5, 6]

After

derived = [1, 1/2, 11, 11/2]

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 ORIGINAL ORIGINAL width over2 interval1 := div(7, 2)

After

Trapezoid ORIGINAL ORIGINAL width over2 interval1 = 7/2

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 ORIGINAL ORIGINAL endpoint sum interval1 := sum(6, 8)

After

Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 = 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, S008

S010 · calculation

Before

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 := mul(7/2, 14)

After

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 = 49

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 ORIGINAL, 1, 1, 8, 6, 8]

After

derived = [7, 7/2, 14, 49]

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 ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 11/2)

After

Trapezoid ORIGINAL ORIGINAL trapezoid running total0 = 11/2

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 ORIGINAL ORIGINAL trapezoid running total1 := sum(11/2, 49)

After

Trapezoid ORIGINAL ORIGINAL trapezoid running total1 = 109/2

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 ORIGINAL, [0, 1, 8], [5, 6, 8]]

After

derived = [[0, 1, 8], [5, 6, 8], [1, 7], [11/2, 49], 109/2]

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

Reverse checks

  • 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 ORIGINAL ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Trapezoid 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 ORIGINAL ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Trapezoid 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 ORIGINAL ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Trapezoid 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

([0, 1, 8], [5, 6, 8], 2, 3, [1, 7], Yes, n_subintervals_have_n_plus_one_nodes)

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: 5725c27cd8685388f9980a92c2c5a8025c791e1a64fd8b1633d9b6ced6a8c23a
  • candidate record SHA-256: 5cf7f8570a74d3e5d985e99d1276f04c2e17960222b5c0606e7e2acde04cc3a7
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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