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R0014-07601 · q-20261012-advance-20261016-r02-matrix-0076
Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], candidate=[[0]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).
G001 — Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], candidate=[[0]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).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
M=[[0]];shape=[1, 1]
After
external indices start at1;transpose is not inverse
Conditions: none
First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion.
Rule: rectangular_matrix_scope · Uses: G001
Before
M transpose:source(1,1)=0
After
target(1,1)=0
Conditions: none
Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based.
Rule: transpose_coordinate_swap · Uses: G001, S001
Before
candidate shape=[1, 1];required transpose shape=[1, 1]
After
qualified=Yes
Conditions: none
The candidate must have the swapped row/column sizes of M.A shape mismatch forbids constructing an entrywise residual.
Rule: candidate_matrix_shape_before_difference · Uses: G001, S002
Before
candidate-MT(1,1) := sub(0, 0)
After
candidate-MT(1,1) = 0
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
qualified=Yes;residual/status=[[0]]
After
equal=Yes
Conditions: none
With matching shapes every entry residual must be zero.Mismatched shapes fail a prerequisite and must not be assigned a fabricated zero residual matrix.
Rule: candidate_matrix_all_entries · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute candidate-MT(1,1) with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([[0]], [1, 1], Yes, [[0]], Yes)
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: 084e4425bf6ebf234475c4c2882430097c37f87fd922a0f3a798feb88e0c3110candidate record SHA-256: 9b9522ea83c524bf1f16eee66aa4a302031ce93fffb0f2553c4ba6a47e8c5c21semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED