# Question 11054: An Exact Point from Qualified Convex Weights [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-0029) 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. R0021-11054 · q-20261013-advance-20261024-r05-finite-convex-combination-0029 ## 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=[[-9, 0], [-7, 2]], weights=[4/6, 2/6]. 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=[[-9, 0], [-7, 2]], weights=[4/6, 2/6]. 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](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/) ## Step-by-step derivation S001 · impliesBefore convex weights:raw rational pairs=[[4, 6], [2, 6]] After convex weights:normalized=[2/3, 1/3] Conditions: none Every original denominator must be nonzero.Reduce fractions to positive denominators without decimal approximation. Step identifiers and dependenciesRule: exact_weight_normalization · Uses: G001 S002 · calculationBefore convex weights total := sum(2/3, 1/3) After convex weights total = 1 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, S001 S003 · impliesBefore convex weights:weights=[2/3, 1/3];sum=1 After nonnegative=Yes;total_one=Yes;qualified=Yes Conditions: none Check nonnegativity and exact total one separately.Zero weights are allowed;negative weights or near-one totals are not repaired. Step identifiers and dependenciesRule: separate_weight_guards · Uses: G001, S002 S004 · impliesBefore convex point:ordered vectors=[[-9, 0], [-7, 2]];aligned weights=[2/3, 1/3] 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, S003 S005 · calculationBefore convex point coordinate0 term0 := mul(2/3, -9) After convex point coordinate0 term0 = -6 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 convex point coordinate0 term1 := mul(1/3, -7) After convex point coordinate0 term1 = -7/3 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 convex point coordinate0 sum := sum(-6, -7/3) After convex point coordinate0 sum = -25/3 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 convex point coordinate1 term0 := mul(2/3, 0) After convex 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, S007 S009 · calculationBefore convex point coordinate1 term1 := mul(1/3, 2) After convex point coordinate1 term1 = 2/3 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 S010 · calculationBefore convex point coordinate1 sum := sum(0, 2/3) After convex point coordinate1 sum = 2/3 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, S009 ## 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 ([[-9, 0], [-7, 2]], [2/3, 1/3], [-25/3, 2/3]) 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: 3639812ddc3d50e20b7a27f8f73832569324e845087095656716c4c8db63a968 - candidate record SHA-256: 3a7aaa97e747cdd84f98915a4f3163722e841f17a9b02b2c9f92fc359c055eb2 - 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-0029.json)