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R0065-33119 · q-20261006-advance-20261007-r01-matrix-0069
Over R, A=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] and B=[[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]] have the same dimensions. Find A+B as a nested list of rows.
G001 — Over R, A=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] and B=[[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]] have the same dimensions. Find A+B as a nested list of rows.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 = [qf-matrix-addition, matrix_addition, reals, []]
After
derived = [[nested_ROW_matrix]]
Conditions: none
MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.
Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001
Before
source roles = [A ORIGINAL, [[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]]]
After
derived = [4, 4]
Conditions: none
THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order.
Rule: Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused · Uses: G001, S001
Before
source roles = [B ORIGINAL, [[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]]]
After
derived = [4, 4]
Conditions: none
THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order.
Rule: Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused · Uses: G001, S002
Before
Matrix sum ORIGINAL entry0,0 := sum(6, -6)
After
Matrix sum ORIGINAL entry0,0 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Matrix sum ORIGINAL entry0,1 := sum(5, -5)
After
Matrix sum ORIGINAL entry0,1 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Matrix sum ORIGINAL entry0,2 := sum(4, -4)
After
Matrix sum ORIGINAL entry0,2 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Matrix sum ORIGINAL entry0,3 := sum(3, -3)
After
Matrix sum ORIGINAL entry0,3 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Matrix sum ORIGINAL entry1,0 := sum(8, -8)
After
Matrix sum ORIGINAL entry1,0 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Matrix sum ORIGINAL entry1,1 := sum(7, -7)
After
Matrix sum ORIGINAL entry1,1 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Matrix sum ORIGINAL entry1,2 := sum(6, -6)
After
Matrix sum ORIGINAL entry1,2 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Matrix sum ORIGINAL entry1,3 := sum(5, -5)
After
Matrix sum ORIGINAL entry1,3 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Matrix sum ORIGINAL entry2,0 := sum(10, -10)
After
Matrix sum ORIGINAL entry2,0 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
Matrix sum ORIGINAL entry2,1 := sum(9, -9)
After
Matrix sum ORIGINAL entry2,1 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Matrix sum ORIGINAL entry2,2 := sum(8, -8)
After
Matrix sum ORIGINAL entry2,2 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
Matrix sum ORIGINAL entry2,3 := sum(7, -7)
After
Matrix sum ORIGINAL entry2,3 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Matrix sum ORIGINAL entry3,0 := sum(12, -12)
After
Matrix sum ORIGINAL entry3,0 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Matrix sum ORIGINAL entry3,1 := sum(11, -11)
After
Matrix sum ORIGINAL entry3,1 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Matrix sum ORIGINAL entry3,2 := sum(10, -10)
After
Matrix sum ORIGINAL entry3,2 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Matrix sum ORIGINAL entry3,3 := sum(9, -9)
After
Matrix sum ORIGINAL entry3,3 = 0
Conditions: none
Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]], [[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]]]
After
derived = [[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]]
Conditions: none
Add EVERY corresponding entry ONLY at equal shapes.Derived sums may exceed SAFE;do not reparse as a RAW matrix.No zero-padding or broadcasting.
Rule: Matrix_ADDITION_all_corresponding_entries_NO_padding_or_raw_recap · Uses: G001, S019
S020 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.S019 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,3 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,2 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,1 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry3,0 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,3 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,2 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry2,0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,3 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,2 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry1,0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,2 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,1 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Matrix sum ORIGINAL entry0,0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
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: 5ece5814775c5e7e48e0f27e5158895d5b8c27c9975a9ef6c2d3040f80a6442fcandidate record SHA-256: 275a7f302bec0fca86f792598fe75624fa56f1dfdfa93aa2e81e34d301841118semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00Lean: NOT_RUN · human review: NOT_PERFORMED