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

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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.

R0021-11012 · q-20261013-advance-20261024-r05-finite-convex-combination-0012



## 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](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/)





## Step-by-step derivation

S001 · impliesBefore

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

After

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

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(6/7, 1/7)

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=[6/7, 1/7];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









## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 2d23496b698a69d2fd096c609737a63fe27d78d4bc25c1ea3696968b764fba63
- candidate record SHA-256: 29ef48b7ee80d8cebdd774b590ecb020a77414f3d08fd9abb04c3c87e49ce2d5
- 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-0012.json)