# Question 25025: Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles [Return to catalogue page 126](https://ocme.evemisslab.com/en/questions/page/126/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations/#q-20261015-advance-20261114-r06-linear-program-standard-form-0100) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0049-25025 · q-20261015-advance-20261114-r06-linear-program-standard-form-0100 ## Original question and givens Use mode=greater_equal, coefficients=[25, -26, 1], rhs=27. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate). - G001 — Use mode=greater_equal, coefficients=[25, -26, 1], rhs=27. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, 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](https://ocme.evemisslab.com/en/mko/mko-linear-program-standard-form/) ## Step-by-step derivation S001 · impliesBefore source roles = [lp-standard-convention-boundary] After derived = [[mode, coefficients, rhs, negated_coefficients, negated_rhs, 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore LP convention coefficient0 negation := mul(-1, 25) After LP convention coefficient0 negation = -25 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore LP convention coefficient1 negation := mul(-1, -26) After LP convention coefficient1 negation = 26 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore LP convention RHS negation := mul(-1, 27) After LP convention RHS negation = -27 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [greater_equal, [25, -26, 1], 27] After derived = [greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal] 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 dependenciesRule: LP_original_mode_roles_and_GENERAL_equivalent_conversion · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute LP convention RHS negation with exact rational arithmetic, not rounded decimals. - 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 (greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal) 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: 382225eb56cf5ca49d746a6d679d2a76b20a9fef70ce422dd5afbde278288c53 - candidate record SHA-256: d1b919a60d39dacc31f684ce5f9c161a96629b165ec1c2504ea6479b54b0a53d - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r06-linear-program-standard-form-0100.json)