Question 11065: An Exact Point from Qualified Convex Weights

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=[[2, 0], [4, 2]], weights=[15/17, 2/17]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). 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=[[2, 0], [4, 2]], weights=[15/17, 2/17]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). 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=[[15, 17], [2, 17]]

convex weights:normalized=[15/17, 2/17]

convex weights total := sum(15/17, 2/17)

convex weights total = 1

convex weights:weights=[15/17, 2/17];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

convex point:ordered vectors=[[2, 0], [4, 2]];aligned weights=[15/17, 2/17]

dimension=2;preserve duplicates and order

convex point coordinate0 term0 := mul(15/17, 2)

convex point coordinate0 term0 = 30/17

convex point coordinate0 term1 := mul(2/17, 4)

convex point coordinate0 term1 = 8/17

convex point coordinate0 sum := sum(30/17, 8/17)

convex point coordinate0 sum = 38/17

convex point coordinate1 term0 := mul(15/17, 0)

convex point coordinate1 term0 = 0

convex point coordinate1 term1 := mul(2/17, 2)

convex point coordinate1 term1 = 4/17

convex point coordinate1 sum := sum(0, 4/17)

convex point coordinate1 sum = 4/17

Reverse checks

• S010 · premise_and_result_audit — Recompute convex point coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute convex point coordinate1 term1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute convex point coordinate1 term0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute convex point coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute convex point coordinate0 term1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute convex point coordinate0 term0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([[2, 0], [4, 2]], [15/17, 2/17], [38/17, 4/17])

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.