Question 11012: Nonnegativity and Exact Total One as Separate Weight Tests

Original question and givens

Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-1, 0], [1, 2]], weights=[12/14, 2/14]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

• G001 — Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-1, 0], [1, 2]], weights=[12/14, 2/14]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

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-finite-convex-combination

Step-by-step derivation

convex weights:raw rational pairs=[[12, 14], [2, 14]]

convex weights:normalized=[6/7, 1/7]

convex weights total := sum(6/7, 1/7)

convex weights total = 1

convex weights:weights=[6/7, 1/7];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

Reverse checks

• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute convex weights total 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, 0], [1, 2]], [6/7, 1/7], Yes, 1, Yes, 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.