Question 11050: Audit lambda without Clipping an Affine Extension

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. left=[12, 0], right=[15, 3], lambda=25/27. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. 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. left=[12, 0], right=[15, 3], lambda=25/27. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. 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

two-point complementary weight := sub(1, 25/27)

two-point complementary weight = 2/27

two-point:raw lambda=[25, 27];lambda=25/27;one-minus-lambda=2/27

in_closed_unit_interval=Yes;lambda weights left

two-point:ordered vectors=[[12, 0], [15, 3]];aligned weights=[25/27, 2/27]

dimension=2;preserve duplicates and order

two-point coordinate0 term0 := mul(25/27, 12)

two-point coordinate0 term0 = 100/9

two-point coordinate0 term1 := mul(2/27, 15)

two-point coordinate0 term1 = 10/9

two-point coordinate0 sum := sum(100/9, 10/9)

two-point coordinate0 sum = 110/9

two-point coordinate1 term0 := mul(25/27, 0)

two-point coordinate1 term0 = 0

two-point coordinate1 term1 := mul(2/27, 3)

two-point coordinate1 term1 = 2/9

two-point coordinate1 sum := sum(0, 2/9)

two-point coordinate1 sum = 2/9

Reverse checks

• S009 · premise_and_result_audit — Recompute two-point coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute two-point coordinate1 term1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute two-point coordinate1 term0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute two-point coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute two-point coordinate0 term1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute two-point coordinate0 term0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute two-point complementary weight with exact rational arithmetic, not rounded decimals.

Result

([12, 0], [15, 3], 25/27, Yes, [110/9, 2/9], 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.