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R0054-27912 · q-20261015-pulse-20261119-r04-0087
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).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-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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization · Uses: G001, S003
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
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.
Rule: Lagrange_rank_EACH_other_nonzero_factor_full_row_elimination · Uses: G001, S008
Before
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.
Rule: Lagrange_rank_no_remaining_pivot_in_column · Uses: G001, S009
Before
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.
Rule: Lagrange_rank_RREF_pivots_equal_row_rank_original_rows_retained · Uses: G001, S010
Before
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.
Rule: Lagrange_ACTUAL_rank_deficiency_NOT_absence_of_extrema · Uses: G001, S011
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.(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.
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: 69c284fe695fe4f5f28d147522c6b311714e218198d75ece92525222a4e8b772candidate record SHA-256: 951c5cf2041f235cdbfc64b9df037d52841ca5ded4a66b6de7db9f6f14883e8asemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED