Question 25082: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split

Original question and givens

Use column=[7, -8, 9], objective coefficient=10, unrestricted value=7. 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=[7, -8, 9], objective coefficient=10, unrestricted value=7. 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(7, 0)

LP unrestricted reconstruct value = 7

LP unrestricted ORIGINAL column0 := mul(7, 7)

LP unrestricted ORIGINAL column0 = 49

LP unrestricted positive column0 := mul(7, 7)

LP unrestricted positive column0 = 49

LP unrestricted negative column0 := mul(-7, 0)

LP unrestricted negative column0 = 0

LP unrestricted split column0 := sum(49, 0)

LP unrestricted split column0 = 49

LP unrestricted ORIGINAL column1 := mul(-8, 7)

LP unrestricted ORIGINAL column1 = -56

LP unrestricted positive column1 := mul(-8, 7)

LP unrestricted positive column1 = -56

LP unrestricted negative column1 := mul(8, 0)

LP unrestricted negative column1 = 0

LP unrestricted split column1 := sum(-56, 0)

LP unrestricted split column1 = -56

LP unrestricted ORIGINAL column2 := mul(9, 7)

LP unrestricted ORIGINAL column2 = 63

LP unrestricted positive column2 := mul(9, 7)

LP unrestricted positive column2 = 63

LP unrestricted negative column2 := mul(-9, 0)

LP unrestricted negative column2 = 0

LP unrestricted split column2 := sum(63, 0)

LP unrestricted split column2 = 63

LP unrestricted ORIGINAL objective contribution := mul(10, 7)

LP unrestricted ORIGINAL objective contribution = 70

LP unrestricted positive objective := mul(10, 7)

LP unrestricted positive objective = 70

LP unrestricted negative objective := mul(-10, 0)

LP unrestricted negative objective = 0

LP unrestricted split objective sum := sum(70, 0)

LP unrestricted split objective sum = 70

source roles = [[7, -8, 9], 10, 7, 7, 0]

derived = [7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, 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

([7, -8, 9], 10, 7, 7, 0, 7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, 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.