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R0054-27953 · q-20261015-pulse-20261119-r04-0003
Use constraint gradients as ordered rows=[[1, 0], [0, 3]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).
G001 — Use constraint gradients as ordered rows=[[1, 0], [0, 3]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).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-regularity-audit]
After
derived = [[constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition]]
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, 3]], 1]
After
derived = [[[1, 0], [0, 3]]]
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, 3)
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(3, 3)
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, 3]], 3]
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, 3]]]
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
source roles = [[[1, 0], [0, 3]]]
After
derived = [2, 2, 2, Yes]
Conditions: none
This qualifies algebraic rank of GIVEN gradient rows only.Without functions/point data it does not independently establish differentiability,feasibility or an extremum.Compare rank with constraint count,not ambient dimension.
Rule: Lagrange_supplied_vector_rank_ONLY_no_unprovided_functions_or_feasibility · Uses: G001, S008
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 — 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.([[1, 0], [0, 3]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)
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: 2208ef839eeb10cc73d17a8269f6e5383b59f27ea77ba0c5e6b969bb56f224facandidate record SHA-256: 9b624250d4b833eeb8012906796932aa30e9c780d2d30b74db2b6d37db1fe04fsemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED