Question 25080: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split

Original question and givens

Use column=[5, -6, 7], objective coefficient=8, unrestricted value=5. 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=[5, -6, 7], objective coefficient=8, unrestricted value=5. 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.

• mko-linear-program-standard-form

Step-by-step derivation

source roles = [lp-standard-unrestricted-split]

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]]

LP unrestricted reconstruct value := sub(5, 0)

LP unrestricted reconstruct value = 5

LP unrestricted ORIGINAL column0 := mul(5, 5)

LP unrestricted ORIGINAL column0 = 25

LP unrestricted positive column0 := mul(5, 5)

LP unrestricted positive column0 = 25

LP unrestricted negative column0 := mul(-5, 0)

LP unrestricted negative column0 = 0

LP unrestricted split column0 := sum(25, 0)

LP unrestricted split column0 = 25

LP unrestricted ORIGINAL column1 := mul(-6, 5)

LP unrestricted ORIGINAL column1 = -30

LP unrestricted positive column1 := mul(-6, 5)

LP unrestricted positive column1 = -30

LP unrestricted negative column1 := mul(6, 0)

LP unrestricted negative column1 = 0

LP unrestricted split column1 := sum(-30, 0)

LP unrestricted split column1 = -30

LP unrestricted ORIGINAL column2 := mul(7, 5)

LP unrestricted ORIGINAL column2 = 35

LP unrestricted positive column2 := mul(7, 5)

LP unrestricted positive column2 = 35

LP unrestricted negative column2 := mul(-7, 0)

LP unrestricted negative column2 = 0

LP unrestricted split column2 := sum(35, 0)

LP unrestricted split column2 = 35

LP unrestricted ORIGINAL objective contribution := mul(8, 5)

LP unrestricted ORIGINAL objective contribution = 40

LP unrestricted positive objective := mul(8, 5)

LP unrestricted positive objective = 40

LP unrestricted negative objective := mul(-8, 0)

LP unrestricted negative objective = 0

LP unrestricted split objective sum := sum(40, 0)

LP unrestricted split objective sum = 40

source roles = [[5, -6, 7], 8, 5, 5, 0]

derived = [5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, Yes]

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

([5, -6, 7], 8, 5, 5, 0, 5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, 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.