# Question 16450: A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure [Return to catalogue page 83](https://ocme.evemisslab.com/en/questions/page/83/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r01-vector-subspace-foundations/#q-20261014-advance-20261101-r01-vector-subspace-0075) 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. R0031-16450 · q-20261014-advance-20261101-r01-vector-subspace-0075 ## Original question and givens Use the complete nonempty duplicate-free finite point set=[[0, 0], [25, 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], [25, 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](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/) ## Step-by-step derivation S001 · impliesBefore DECLARED complete finite candidate subset of usual real R2=[[0, 0], [25, 1]];original point order retained After zero vector present=Yes;raw points nonempty and duplicate-free=Yes Conditions: none The complete finite list is the candidate set itself,not a sample from an unknown subspace.Keep original point order and prefer a missing-zero counterexample;do not deduplicate or reorder to change the source witness. Step identifiers and dependenciesRule: subspace_complete_finite_set_not_unknown_sampled_universe · Uses: G001 S002 · impliesBefore first original nonzero point index=1;point=[25, 1];complete set size=2 After integer multiples k*point for k=2 through4 are 3 DISTINCT vectors;some image must be absent Conditions: none A nonzero coordinate lets equal scalar multiples imply equal scalars.Thus L+1 distinct multiples cannot all lie in an L-point set.This guarantees a missing multiple rather than guessing closure from positive tests. Step identifiers and dependenciesRule: subspace_nonzero_real_multiples_finite_pigeonhole_obligation · Uses: G001, S001 S003 · calculationBefore finite scalar2 image coordinate0 := mul(2, 25) After finite scalar2 image coordinate0 = 50 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore finite scalar2 image coordinate1 := mul(2, 1) After finite scalar2 image coordinate1 = 2 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore first original nonzero point=[25, 1];candidate scalar=2;exact image=[50, 2] After image belongs to complete candidate=No Conditions: none Test actual membership for scalars2,3,...in order.The first missing image is the source witness.Derived coordinates may exceed raw input bounds;that does not erase the counterexample. Step identifiers and dependenciesRule: subspace_first_missing_integer_scalar_in_source_order · Uses: G001, S004 ## 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], [25, 1]], [scalar_failure, 2, [25, 1], [50, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 12be1e2eefe5ae21d8d18bf196c9fd870d63f5247d3d02c899684aa45f6ae410 - candidate record SHA-256: 1ae2aed52ba138b4b900b2b0493f542e7b4ffb2e13c7356d0cd50d8db5d002b8 - semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a - question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r01-vector-subspace-0075.json)