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R0070-35936 · q-20261003-advance-20261004-r05-vector-0061
In the fixed basis of the real plane R², u=(-2, -2) and v=(0, 2). Compute the exact components of u−v, preserving the subtraction order.
G001 — In the fixed basis of the real plane R², u=(-2, -2) and v=(0, 2). Compute the exact components of u−v, preserving the subtraction order.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-difference, [domain, operation, u, v]]
After
derived = [vector, vector_subtract, 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], [0, 2], 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 := sub(-2, 0)
After
Vector ORIGINAL coordinate i=0 = -2
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 := sub(-2, 2)
After
Vector ORIGINAL coordinate i=1 = -4
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], [0, 2], null, vector_subtract]
After
derived = [[-2, -4], (-2, -4)]
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.(-2, -4)
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: 20c70fc472f014fb4a1905500d3f088de6045420d48d1d5eb4bf3908f13d0779candidate record SHA-256: 40516d21b666d0fa9bb7afc05d0abf87c8207ded43fd6cd39ebb4b4a7c46e413semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dabLean: NOT_RUN · human review: NOT_PERFORMED