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R0070-35915 · q-20261003-advance-20261004-r05-vector-0015
In the fixed basis of the real plane R², u=(2, 2) and v=(-2, -3). Compute the exact components of u+v.
G001 — In the fixed basis of the real plane R², u=(2, 2) and v=(-2, -3). Compute the exact components of u+v.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-vector-coordinate-addition, [domain, operation, u, v]]
After
derived = [vector, vector_add, reals]
Conditions: none
Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.
Rule: R70_SOURCE_bank_specific_interface · Uses: G001
Before
source roles = [[2, 2], [-2, -3], null]
After
derived = [two real coordinates in ONE ordered basis]
Conditions: none
The vectors use ONE ordered basis.Type ALL TWO RAW SAFE components and k.These are vector coordinates,not points,lengths or a dot product.Component sums alone do not determine equality.
Rule: Vector_GIVEN_common_ordered_basis_RAW_two_SAFE_components · Uses: G001, S001
Before
Vector ORIGINAL coordinate i=0 := sum(2, -2)
After
Vector ORIGINAL coordinate i=0 = 0
Conditions: none
This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Vector ORIGINAL coordinate i=1 := sum(2, -3)
After
Vector ORIGINAL coordinate i=1 = -1
Conditions: none
This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [[2, 2], [-2, -3], null, vector_add]
After
derived = [[0, -1], (0, -1)]
Conditions: none
Operate componentwise in ORIGINAL order.u−v must not reverse;zero scaling yields the zero vector.The result is an ORDERED tuple,not a set or length.Derived components may exceed RAW SAFE.
Rule: Vector_ORDERED_componentwise_result · Uses: G001, S004
S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S004 · premise_and_result_audit — Recompute Vector ORIGINAL coordinate i=1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Vector ORIGINAL coordinate i=0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.(0, -1)
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: 1689473bf275671a11d22fb22f1972e47282c957f04a1f3ffba801a4e0001956candidate record SHA-256: 1a182329c2784a0d5481190208ee34aaec85d13e2857b523b4e52bd80d369d66semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dabLean: NOT_RUN · human review: NOT_PERFORMED