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R0049-25019 · q-20261015-advance-20261114-r06-linear-program-standard-form-0094
Use mode=greater_equal, coefficients=[19, -20, 1], rhs=21. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
G001 — Use mode=greater_equal, coefficients=[19, -20, 1], rhs=21. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).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.
Before
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
LP convention coefficient0 negation := mul(-1, 19)
After
LP convention coefficient0 negation = -19
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
LP convention coefficient1 negation := mul(-1, -20)
After
LP convention coefficient1 negation = 20
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S002
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
LP convention RHS negation := mul(-1, 21)
After
LP convention RHS negation = -21
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [greater_equal, [19, -20, 1], 21]
After
derived = [greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, 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.
Rule: LP_original_mode_roles_and_GENERAL_equivalent_conversion · Uses: G001, S005
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.(greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f72ae047874132dbe80b3d5d4a4a4016fdc08127d6c7cee6ea4cb9edbaffced8candidate record SHA-256: 5ca2905be3a11bd49e7055f61d7b63c1cf79d8ff00237706a4d280dd9778e1ddsemantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED