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R0014-07920 · q-20261012-advance-20261016-r01-vector-0095
Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 2], a=2, b=1, candidate=[16, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).
G001 — Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 2], a=2, b=1, candidate=[16, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).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
fixed standard basis;two ordered integer coordinates
After
componentwise operations;not length or inner product
Conditions: none
Work coordinate by coordinate in the same fixed basis.Swapping coordinates,taking a norm or computing an inner product is not the requested vector answer.
Rule: ordered_vector_scope · Uses: G001
Before
au[0] := mul(2, 7)
After
au[0] = 14
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
au[1] := mul(2, 1)
After
au[1] = 2
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
bv[0] := mul(1, 1)
After
bv[0] = 1
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
bv[1] := mul(1, 2)
After
bv[1] = 2
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
au+bv[0] := sum(14, 1)
After
au+bv[0] = 15
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
au+bv[1] := sum(2, 2)
After
au+bv[1] = 4
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
au=[14, 2];bv=[1, 2]
After
value=[15, 4]
Conditions: none
Each weight multiplies its own vector before aligned coordinates are added.Negative weights and the zero vector use the same definition.
Rule: weighted_vector_components · Uses: G001, S007
Before
candidate-value[0] := sub(16, 15)
After
candidate-value[0] = 1
Conditions: none
Subtract the second operand from the first in the requested order,retaining the residual sign.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
candidate-value[1] := sub(4, 4)
After
candidate-value[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, S009
Before
candidate=[16, 4];residual=[1, 0]
After
equal=No
Conditions: none
Both components of candidate minus true value must be zero.One matching coordinate does not establish equality of the entire vector.
Rule: vector_candidate_all_coordinates · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute candidate-value[1] with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute candidate-value[0] with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute au+bv[1] with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute au+bv[0] with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute bv[1] with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute bv[0] with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute au[1] with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute au[0] with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([15, 4], [1, 0], No)
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: ae070d8e4ccbabb00fb0cc891e1ac05c94a1d6df2bf2073bff1e55f3e948930ecandidate record SHA-256: b648d879e2fc6378d3f78eafa91a3c4722633fc3c35d1ebabf1efb8ede883135semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED