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Question 25081: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split

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

R0049-25081 · q-20261015-advance-20261114-r06-linear-program-standard-form-0056

Original question and givens

Use column=[6, -7, 8], objective coefficient=9, unrestricted value=-6. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).

  • G001 — Use column=[6, -7, 8], objective coefficient=9, unrestricted value=-6. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).

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 = [lp-standard-unrestricted-split]

After

derived = [[column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

LP unrestricted reconstruct value := sub(0, 6)

After

LP unrestricted reconstruct value = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

LP unrestricted ORIGINAL column0 := mul(6, -6)

After

LP unrestricted ORIGINAL column0 = -36

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

LP unrestricted positive column0 := mul(6, 0)

After

LP unrestricted positive column0 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

LP unrestricted negative column0 := mul(-6, 6)

After

LP unrestricted negative column0 = -36

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

LP unrestricted split column0 := sum(0, -36)

After

LP unrestricted split column0 = -36

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

LP unrestricted ORIGINAL column1 := mul(-7, -6)

After

LP unrestricted ORIGINAL column1 = 42

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

LP unrestricted positive column1 := mul(-7, 0)

After

LP unrestricted positive column1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

LP unrestricted negative column1 := mul(7, 6)

After

LP unrestricted negative column1 = 42

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

LP unrestricted split column1 := sum(0, 42)

After

LP unrestricted split column1 = 42

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

LP unrestricted ORIGINAL column2 := mul(8, -6)

After

LP unrestricted ORIGINAL column2 = -48

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

LP unrestricted positive column2 := mul(8, 0)

After

LP unrestricted positive column2 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

LP unrestricted negative column2 := mul(-8, 6)

After

LP unrestricted negative column2 = -48

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

LP unrestricted split column2 := sum(0, -48)

After

LP unrestricted split column2 = -48

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

LP unrestricted ORIGINAL objective contribution := mul(9, -6)

After

LP unrestricted ORIGINAL objective contribution = -54

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

LP unrestricted positive objective := mul(9, 0)

After

LP unrestricted positive objective = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

LP unrestricted negative objective := mul(-9, 6)

After

LP unrestricted negative objective = -54

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

LP unrestricted split objective sum := sum(0, -54)

After

LP unrestricted split objective sum = -54

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [[6, -7, 8], 9, -6, 0, 6]

After

derived = [-6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes]

Conditions: none

Both parts are nonnegative and v=plus−minus.Each row satisfies a*v=a*plus+(−a)*minus,and so does the objective.Projection back is valid,but adding any s>=0 to both parts gives the same v,so this is not a bijection of all split pairs.

Step identifiers and dependencies

Rule: LP_nonnegative_split_preserves_contributions_NOT_all_pair_bijection · Uses: G001, S018

Reverse checks

  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Recompute LP unrestricted split objective sum with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute LP unrestricted negative objective with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute LP unrestricted positive objective with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL objective contribution with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute LP unrestricted split column2 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute LP unrestricted negative column2 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute LP unrestricted positive column2 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column2 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute LP unrestricted split column1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute LP unrestricted negative column1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute LP unrestricted positive column1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column1 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute LP unrestricted split column0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute LP unrestricted negative column0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute LP unrestricted positive column0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute LP unrestricted reconstruct value with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([6, -7, 8], 9, -6, 0, 6, -6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)

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: ce04cbc34ec31638348d588f727f04de992fc767f51753723099eb943ea9904f
  • candidate record SHA-256: 949fdde37b976dbae1971c942869507786cc6293d2da2fee4f576cb2f3aaddb0
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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