Question 16430: A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Original question and givens

Use the complete nonempty duplicate-free finite point set=[[0, 0], [5, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

• G001 — Use the complete nonempty duplicate-free finite point set=[[0, 0], [5, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

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-subspace

Step-by-step derivation

DECLARED complete finite candidate subset of usual real R2=[[0, 0], [5, 1]];original point order retained

zero vector present=Yes;raw points nonempty and duplicate-free=Yes

first original nonzero point index=1;point=[5, 1];complete set size=2

integer multiples k*point for k=2 through4 are 3 DISTINCT vectors;some image must be absent

finite scalar2 image coordinate0 := mul(2, 5)

finite scalar2 image coordinate0 = 10

finite scalar2 image coordinate1 := mul(2, 1)

finite scalar2 image coordinate1 = 2

first original nonzero point=[5, 1];candidate scalar=2;exact image=[10, 2]

image belongs to complete candidate=No

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute finite scalar2 image coordinate1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite scalar2 image coordinate0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], No, not_subspace_exact_counterexample)

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.