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R0058-29605 · q-20261013-advance-20261022-r04-real-vector-space-0005
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-8, 1, -2], right=[2, -4, 1]. Give (left,right,the componentwise sum).
G001 — Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-8, 1, -2], right=[2, -4, 1]. Give (left,right,the componentwise sum).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-str22-vectorspace-addition, vectorspace-addition]
After
derived = [[left, right, result]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
Vector ORIGINAL addition coordinate0 := sum(-8, 2)
After
Vector ORIGINAL addition coordinate0 = -6
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Vector ORIGINAL addition coordinate1 := sum(1, -4)
After
Vector ORIGINAL addition coordinate1 = -3
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Vector ORIGINAL addition coordinate2 := sum(-2, 1)
After
Vector ORIGINAL addition coordinate2 = -1
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [[-8, 1, -2], [2, -4, 1]]
After
derived = [[-6, -3, -1]]
Conditions: none
Add matching coordinates in the fixed R^d and require equal nonempty dimensions.Derived coordinates are not clipped to raw1000.This is not a proof of all axioms for arbitrary abstract spaces.
Rule: Vector_addition_EVERY_matching_coordinate_same_dimension · Uses: G001, S004
S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S004 · premise_and_result_audit — Recompute Vector ORIGINAL addition coordinate2 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Vector ORIGINAL addition coordinate1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Vector ORIGINAL addition coordinate0 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.([-8, 1, -2], [2, -4, 1], [-6, -3, -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: 2b9c09d6657bdf182780f89a44077dd5e1c27628134f647a14e8b8e305cc5403candidate record SHA-256: 0fbf1384032ab41076b962a976ff679b41d268e307b571d8b438e9a5497dc4f3semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED