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Question 25015: Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles

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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R0049-25015 · q-20261015-advance-20261114-r06-linear-program-standard-form-0090

Original question and givens

Use mode=minimize, coefficients=[15, -16, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).

  • G001 — Use mode=minimize, coefficients=[15, -16, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).

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-convention-boundary]

After

derived = [[mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary]]

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 convention coefficient0 negation := mul(-1, 15)

After

LP convention coefficient0 negation = -15

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 convention coefficient1 negation := mul(-1, -16)

After

LP convention coefficient1 negation = 16

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 convention coefficient2 negation := mul(-1, 1)

After

LP convention coefficient2 negation = -1

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 · implies

Before

source roles = [minimize, [15, -16, 1], 0]

After

derived = [minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal]

Conditions: none

Multiplying a >=constraint by−1 reverses its sign;equality is two opposite <=constraints;minimizing cTx is maximizing(−c)Tx with negated value.For minimize,rhs0 is only an unused-field sentinel.Conversion preserves pointwise constraints/objectives,not feasibility,boundedness,existence of an optimum or duality;no b>=0 assumption.

Step identifiers and dependencies

Rule: LP_original_mode_roles_and_GENERAL_equivalent_conversion · Uses: G001, S004

Reverse checks

  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Recompute LP convention coefficient2 negation with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute LP convention coefficient1 negation with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute LP convention coefficient0 negation 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

(minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)

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: 9690e16ed474da5ca2fb6b5a7cd14b0315f14994098c3479a76baa5026bf6e76
  • candidate record SHA-256: fd8495ca3db4e66e106114ab0e2f40cd274342cf87b73f19f47e1759be452666
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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