# Question 7903: Weighted Vector Candidate and Ordered Residual

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0014-07903 · q-20261012-advance-20261016-r01-vector-0078



## Original question and givens

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=[1, 2], v=[1, 2], a=2, b=-1, candidate=[1, 2]. 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=[1, 2], v=[1, 2], a=2, b=-1, candidate=[1, 2]. Give (au+bv, candidate−actual value, is the candidate equal?).





## Definition and scope

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.


- [mko-vector](https://ocme.evemisslab.com/en/mko/mko-vector/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: ordered_vector_scope · Uses: G001





S002 · calculationBefore

au[0] := mul(2, 1)

After

au[0] = 2

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

au[1] := mul(2, 2)

After

au[1] = 4

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

au+bv[0] := sum(2, -1)

After

au+bv[0] = 1

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

au+bv[1] := sum(4, -2)

After

au+bv[1] = 2

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

au=[2, 4];bv=[-1, -2]

After

value=[1, 2]

Conditions: none

Each weight multiplies its own vector before aligned coordinates are added.Negative weights and the zero vector use the same definition.

Step identifiers and dependenciesRule: weighted_vector_components · Uses: G001, S007





S009 · calculationBefore

candidate-value[0] := sub(1, 1)

After

candidate-value[0] = 0

Conditions: none

Subtract the second operand from the first in the requested order,retaining the residual sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

candidate-value[1] := sub(2, 2)

After

candidate-value[1] = 0

Conditions: none

Subtract the second operand from the first in the requested order,retaining the residual sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

candidate=[1, 2];residual=[0, 0]

After

equal=Yes

Conditions: none

Both components of candidate minus true value must be zero.One matching coordinate does not establish equality of the entire vector.

Step identifiers and dependenciesRule: vector_candidate_all_coordinates · Uses: G001, S010









## Reverse checks


- 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.





## Result

([1, 2], [0, 0], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 4da9738cb984cdc946a588351ab51b26669768363de1e4e12d9b182cca1cfeac
- candidate record SHA-256: f551d87bffe2e0b0cdfe39a5fda421efda84cac7a829c87a5ab56310a924740e
- semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
- question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r01-vector-0078.json)