# Question 27938: The Multiplier Equation Uses a Fixed Positive-Sign Convention [Return to catalogue page 140](https://ocme.evemisslab.com/en/questions/page/140/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations/#q-20261015-pulse-20261119-r04-0038) 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. R0054-27938 · q-20261015-pulse-20261119-r04-0038 ## Original question and givens Use objective gradient=[13, 26], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[13, 26]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention). - G001 — Use objective gradient=[13, 26], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[13, 26]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention). ## 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-lagrange-multiplier-condition](https://ocme.evemisslab.com/en/mko/mko-lagrange-multiplier-condition/) ## Step-by-step derivation S001 · impliesBefore source roles = [lagrange-multiplier-residual] After derived = [[objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Lagrange ORIGINAL ORIGINAL normalize row0 col0 := div(1, 1) After Lagrange ORIGINAL ORIGINAL normalize row0 col0 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Lagrange ORIGINAL ORIGINAL normalize row0 col1 := div(0, 1) After Lagrange ORIGINAL ORIGINAL normalize row0 col1 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Lagrange ORIGINAL, 0, 0, 0, [[1, 0], [0, 1]], 1] After derived = [[[1, 0], [0, 1]]] Conditions: none Use the original first-nonzero pivot order and swap rows.Divide only by a nonzero pivot and normalize EVERY column;the inverse operation multiplies back,so row rank is preserved.All unreduced source1e8 intermediates remain qualified. Step identifiers and dependenciesRule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S003 S005 · calculationBefore Lagrange ORIGINAL ORIGINAL normalize row1 col0 := div(0, 1) After Lagrange ORIGINAL ORIGINAL normalize row1 col0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Lagrange ORIGINAL ORIGINAL normalize row1 col1 := div(1, 1) After Lagrange ORIGINAL ORIGINAL normalize row1 col1 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Lagrange ORIGINAL, 1, 1, 1, [[1, 0], [0, 1]], 1] After derived = [[[1, 0], [0, 1]]] Conditions: none Use the original first-nonzero pivot order and swap rows.Divide only by a nonzero pivot and normalize EVERY column;the inverse operation multiplies back,so row rank is preserved.All unreduced source1e8 intermediates remain qualified. Step identifiers and dependenciesRule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S006 S008 · impliesBefore source roles = [Lagrange ORIGINAL, [[1, 0], [0, 1]]] After derived = [[[1, 0], [0, 1]], [[0, 0], [1, 1]], 2] Conditions: none Distinct leading1 pivots make the nonzero rows independent;remaining rows are zero and operations are invertible,so pivot count equals original row rank.Retain ORIGINAL constraint rows for J_g/lambda pairing,not RREF substitutes. Step identifiers and dependenciesRule: Lagrange_rank_RREF_pivots_equal_row_rank_original_rows_retained · Uses: G001, S007 S009 · calculationBefore Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product0 := mul(1, 13) After Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product0 = 13 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum0 := sum(0, 13) After Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum0 = 13 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product1 := mul(0, 26) After Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum1 := sum(13, 0) After Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum1 = 13 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product0 := mul(0, 13) After Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum0 := sum(0, 0) After Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product1 := mul(1, 26) After Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product1 = 26 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum1 := sum(0, 26) After Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum1 = 26 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Lagrange ORIGINAL residual coordinate0 := sub(13, 13) After Lagrange ORIGINAL residual coordinate0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Lagrange ORIGINAL residual coordinate1 := sub(26, 26) After Lagrange ORIGINAL residual coordinate1 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [[13, 26], [[1, 0], [0, 1]], [13, 26]] After derived = [[13, 26], [0, 0], Yes] Conditions: none Pair ORIGINAL rows with their aligned lambdas to compute J_g^T lambda.The convention is grad f equals that combination;do not silently change signs.Every residual must vanish for this algebraic stationarity check.Given vectors do not establish missing feasibility,differentiability or a minimum. Step identifiers and dependenciesRule: Lagrange_ORIGINAL_row_lambda_alignment_POS_sign_complete_residual · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S018 · premise_and_result_audit — Recompute Lagrange ORIGINAL residual coordinate1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Lagrange ORIGINAL residual coordinate0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S006 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S003 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. ## Result ([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda) 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: 65dc27b55609fb68d66a8d172220743fbe907d1b59284a38e3b6decdaa7d7fd0 - candidate record SHA-256: 2ae5ca6ca7f630a30107868f9746544070cca2c8e523fd340965444ec1a8f924 - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r04-0038.json)