# Question 11043: Audit lambda without Clipping an Affine Extension [Return to catalogue page 56](https://ocme.evemisslab.com/en/questions/page/56/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r05-finite-convex-combination-foundations/#q-20261013-advance-20261024-r05-finite-convex-combination-0068) 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. R0021-11043 · q-20261013-advance-20261024-r05-finite-convex-combination-0068 ## 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=[5, 0], right=[8, 3], lambda=18/20. 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=[5, 0], right=[8, 3], lambda=18/20. 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](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/) ## Step-by-step derivation S001 · calculationBefore two-point complementary weight := sub(1, 9/10) After two-point complementary weight = 1/10 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore two-point:raw lambda=[18, 20];lambda=9/10;one-minus-lambda=1/10 After in_closed_unit_interval=Yes;lambda weights left Conditions: none Lambda weights the left endpoint and1−lambda the right.Lambda0 selects right,1 selects left.Outside values still define the affine expression,but not a convex combination;do not clip. Step identifiers and dependenciesRule: two_point_direction_and_qualification · Uses: G001, S001 S003 · impliesBefore two-point:ordered vectors=[[5, 0], [8, 3]];aligned weights=[9/10, 1/10] After dimension=2;preserve duplicates and order Conditions: none Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate. Step identifiers and dependenciesRule: ordered_vector_weight_pairing · Uses: G001, S002 S004 · calculationBefore two-point coordinate0 term0 := mul(9/10, 5) After two-point coordinate0 term0 = 9/2 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore two-point coordinate0 term1 := mul(1/10, 8) After two-point coordinate0 term1 = 4/5 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore two-point coordinate0 sum := sum(9/2, 4/5) After two-point coordinate0 sum = 53/10 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore two-point coordinate1 term0 := mul(9/10, 0) After two-point coordinate1 term0 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore two-point coordinate1 term1 := mul(1/10, 3) After two-point coordinate1 term1 = 3/10 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore two-point coordinate1 sum := sum(0, 3/10) After two-point coordinate1 sum = 3/10 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 ## 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 ([5, 0], [8, 3], 9/10, Yes, [53/10, 3/10], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: f4655daaed252c75041d372c0818ff62de4605a0d7f487285d30653ca0c55310 - candidate record SHA-256: 1c7278dc3fdcc71bb42d17249d9fe47a4beed0ca7611185ad80561463e091e1c - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r05-finite-convex-combination-0068.json)