Question 25078: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split

Original question and givens

Use column=[3, -4, 5], objective coefficient=6, unrestricted value=3. 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=[3, -4, 5], objective coefficient=6, unrestricted value=3. 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(3, 0)

LP unrestricted reconstruct value = 3

LP unrestricted ORIGINAL column0 := mul(3, 3)

LP unrestricted ORIGINAL column0 = 9

LP unrestricted positive column0 := mul(3, 3)

LP unrestricted positive column0 = 9

LP unrestricted negative column0 := mul(-3, 0)

LP unrestricted negative column0 = 0

LP unrestricted split column0 := sum(9, 0)

LP unrestricted split column0 = 9

LP unrestricted ORIGINAL column1 := mul(-4, 3)

LP unrestricted ORIGINAL column1 = -12

LP unrestricted positive column1 := mul(-4, 3)

LP unrestricted positive column1 = -12

LP unrestricted negative column1 := mul(4, 0)

LP unrestricted negative column1 = 0

LP unrestricted split column1 := sum(-12, 0)

LP unrestricted split column1 = -12

LP unrestricted ORIGINAL column2 := mul(5, 3)

LP unrestricted ORIGINAL column2 = 15

LP unrestricted positive column2 := mul(5, 3)

LP unrestricted positive column2 = 15

LP unrestricted negative column2 := mul(-5, 0)

LP unrestricted negative column2 = 0

LP unrestricted split column2 := sum(15, 0)

LP unrestricted split column2 = 15

LP unrestricted ORIGINAL objective contribution := mul(6, 3)

LP unrestricted ORIGINAL objective contribution = 18

LP unrestricted positive objective := mul(6, 3)

LP unrestricted positive objective = 18

LP unrestricted negative objective := mul(-6, 0)

LP unrestricted negative objective = 0

LP unrestricted split objective sum := sum(18, 0)

LP unrestricted split objective sum = 18

source roles = [[3, -4, 5], 6, 3, 3, 0]

derived = [3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, 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

([3, -4, 5], 6, 3, 3, 0, 3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, 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.