# Question 27912: Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT [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-r04-0087) 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. R0054-27912 · q-20261015-pulse-20261119-r04-0087 ## Original question and givens Use mode=rank_deficient, complete model={"constraint_gradients":[[1,12],[2,24]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope). - G001 — Use mode=rank_deficient, complete model={"constraint_gradients":[[1,12],[2,24]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope). ## 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-boundary] After derived = [[mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope]] 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 boundary ORIGINAL ORIGINAL normalize row0 col0 := div(1, 1) After Lagrange boundary 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 boundary ORIGINAL ORIGINAL normalize row0 col1 := div(12, 1) After Lagrange boundary ORIGINAL ORIGINAL normalize row0 col1 = 12 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 boundary ORIGINAL, 0, 0, 0, [[1, 12], [2, 24]], 1] After derived = [[[1, 12], [2, 24]]] 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 boundary ORIGINAL ORIGINAL elimination row1 col0 factor := mul(2, 1) After Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor = 2 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 boundary ORIGINAL ORIGINAL elimination row1 col0 subtract := sub(2, 2) After Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract = 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, S005 S007 · calculationBefore Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor := mul(2, 12) After Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor = 24 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, S006 S008 · calculationBefore Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract := sub(24, 24) After Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract = 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, S007 S009 · impliesBefore source roles = [Lagrange boundary ORIGINAL, 0, 0, 1, 2, [2, 24]] After derived = [[[1, 12], [0, 0]]] Conditions: none Take the factor from the pre-update row;subtract the SAME factor times the pivot row from every column.This full-row operation is invertible.Skip zero factors as the source does;row operations do not redefine original constraint order. Step identifiers and dependenciesRule: Lagrange_rank_EACH_other_nonzero_factor_full_row_elimination · Uses: G001, S008 S010 · impliesBefore source roles = [Lagrange boundary ORIGINAL, 1, 1, [[1, 12], [0, 0]]] After derived = [null] Conditions: none All remaining rows are zero in this column,so it adds no rank;continue to later columns. Step identifiers and dependenciesRule: Lagrange_rank_no_remaining_pivot_in_column · Uses: G001, S009 S011 · impliesBefore source roles = [Lagrange boundary ORIGINAL, [[1, 12], [2, 24]]] After derived = [[[1, 12], [0, 0]], [[0, 0]], 1] 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, S010 S012 · impliesBefore source roles = [[[1, 12], [2, 24]], 1, 2] After derived = [No] Conditions: none Rank below the constraint count withholds regular full-row-rank qualification only;it does not imply no extrema or invalidate every multiplier concept. Step identifiers and dependenciesRule: Lagrange_ACTUAL_rank_deficiency_NOT_absence_of_extrema · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S008 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor 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 boundary ORIGINAL ORIGINAL normalize row0 col1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Lagrange boundary 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 (rank_deficient, [[1, 12], [2, 24]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum) 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: 69c284fe695fe4f5f28d147522c6b311714e218198d75ece92525222a4e8b772 - candidate record SHA-256: 951c5cf2041f235cdbfc64b9df037d52841ca5ded4a66b6de7db9f6f14883e8a - 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-r04-0087.json)