Explore related knowledge and questions →
← Question derivation index · Original question batch
R0054-27946 · q-20261015-pulse-20261119-r04-0046
Use objective gradient=[21, 42], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[21, 42]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).
G001 — Use objective gradient=[21, 42], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[21, 42]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).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 = [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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S003
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S006
Before
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.
Rule: Lagrange_rank_RREF_pivots_equal_row_rank_original_rows_retained · Uses: G001, S007
Before
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product0 := mul(1, 21)
After
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product0 = 21
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum0 := sum(0, 21)
After
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum0 = 21
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL product1 := mul(0, 42)
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum1 := sum(21, 0)
After
Lagrange ORIGINAL transpose combination coordinate0 ORIGINAL running sum1 = 21
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product0 := mul(0, 21)
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product1 := mul(1, 42)
After
Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL product1 = 42
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum1 := sum(0, 42)
After
Lagrange ORIGINAL transpose combination coordinate1 ORIGINAL running sum1 = 42
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Lagrange ORIGINAL residual coordinate0 := sub(21, 21)
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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Lagrange ORIGINAL residual coordinate1 := sub(42, 42)
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.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
source roles = [[21, 42], [[1, 0], [0, 1]], [21, 42]]
After
derived = [[21, 42], [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.
Rule: Lagrange_ORIGINAL_row_lambda_alignment_POS_sign_complete_residual · Uses: G001, S018
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.([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [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.
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: 0f28f76080cb3858b76f5dc14e31997f3db5d2632aa95c4c7f63ce536acd64c0candidate record SHA-256: 3dd3d18011e2038cdb8616da14de52c7c51b310ef13f0ae34d1e8e1260a0f1f9semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED