Exact Convex Weights, Weighted Points and Witnesses
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batch-20261013-advance-20261024-r05-finite-convex-combination-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-12, 0], [-10, 2]], weights=[2/3, 2/3]. 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.
View answer and explanation
([[-12, 0], [-10, 2]], [2/3, 2/3], Yes, 4/3, No, No)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-12, 0], [-10, 2]], [2/3, 2/3], Yes, 4/3, No, No).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
2. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-11, 0], [-9, 2]], weights=[2/4, 2/4]. 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.
View answer and explanation
([[-11, 0], [-9, 2]], [1/2, 1/2], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-11, 0], [-9, 2]], [1/2, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
3. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-10, 0], [-8, 2]], weights=[3/5, 2/5]. 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.
View answer and explanation
([[-10, 0], [-8, 2]], [3/5, 2/5], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-10, 0], [-8, 2]], [3/5, 2/5], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
4. Nonnegativity and Exact Total One as Separate Weight Tests
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]. 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.
View answer and explanation
([[-9, 0], [-7, 2]], [2/3, 1/3], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-9, 0], [-7, 2]], [2/3, 1/3], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
5. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-8, 0], [-6, 2]], weights=[5/7, 2/7]. 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.
View answer and explanation
([[-8, 0], [-6, 2]], [5/7, 2/7], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-8, 0], [-6, 2]], [5/7, 2/7], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
6. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-7, 0], [-5, 2]], weights=[2/3, 2/3]. 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.
View answer and explanation
([[-7, 0], [-5, 2]], [2/3, 2/3], Yes, 4/3, No, No)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-7, 0], [-5, 2]], [2/3, 2/3], Yes, 4/3, No, No).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
7. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-6, 0], [-4, 2]], weights=[7/9, 2/9]. 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.
View answer and explanation
([[-6, 0], [-4, 2]], [7/9, 2/9], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-6, 0], [-4, 2]], [7/9, 2/9], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
8. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-5, 0], [-3, 2]], weights=[8/10, 2/10]. 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.
View answer and explanation
([[-5, 0], [-3, 2]], [4/5, 1/5], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-5, 0], [-3, 2]], [4/5, 1/5], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
9. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-4, 0], [-2, 2]], weights=[9/11, 2/11]. 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.
View answer and explanation
([[-4, 0], [-2, 2]], [9/11, 2/11], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-4, 0], [-2, 2]], [9/11, 2/11], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
10. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[-3, 0], [-1, 2]], weights=[10/12, 2/12]. 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.
View answer and explanation
([[-3, 0], [-1, 2]], [5/6, 1/6], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-3, 0], [-1, 2]], [5/6, 1/6], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
11. Nonnegativity and Exact Total One as Separate Weight Tests
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], [0, 2]], weights=[2/3, 2/3]. 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.
View answer and explanation
([[-2, 0], [0, 2]], [2/3, 2/3], Yes, 4/3, No, No)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-2, 0], [0, 2]], [2/3, 2/3], Yes, 4/3, No, No).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
12. Nonnegativity and Exact Total One as Separate Weight Tests
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.
View answer and explanation
([[-1, 0], [1, 2]], [6/7, 1/7], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-1, 0], [1, 2]], [6/7, 1/7], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
13. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[0, 0], [2, 2]], weights=[13/15, 2/15]. 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.
View answer and explanation
([[0, 0], [2, 2]], [13/15, 2/15], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 2]], [13/15, 2/15], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
14. Nonnegativity and Exact Total One as Separate Weight Tests
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], [3, 2]], weights=[14/16, 2/16]. 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.
View answer and explanation
([[1, 0], [3, 2]], [7/8, 1/8], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 2]], [7/8, 1/8], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
15. Nonnegativity and Exact Total One as Separate Weight Tests
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]. 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.
View answer and explanation
([[2, 0], [4, 2]], [15/17, 2/17], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 0], [4, 2]], [15/17, 2/17], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
16. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[3, 0], [5, 2]], weights=[2/3, 2/3]. 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.
View answer and explanation
([[3, 0], [5, 2]], [2/3, 2/3], Yes, 4/3, No, No)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 0], [5, 2]], [2/3, 2/3], Yes, 4/3, No, No).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
17. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[4, 0], [6, 2]], weights=[17/19, 2/19]. 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.
View answer and explanation
([[4, 0], [6, 2]], [17/19, 2/19], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 0], [6, 2]], [17/19, 2/19], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
18. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[5, 0], [7, 2]], weights=[18/20, 2/20]. 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.
View answer and explanation
([[5, 0], [7, 2]], [9/10, 1/10], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 0], [7, 2]], [9/10, 1/10], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
19. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[6, 0], [8, 2]], weights=[19/21, 2/21]. 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.
View answer and explanation
([[6, 0], [8, 2]], [19/21, 2/21], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 0], [8, 2]], [19/21, 2/21], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
20. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[7, 0], [9, 2]], weights=[20/22, 2/22]. 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.
View answer and explanation
([[7, 0], [9, 2]], [10/11, 1/11], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 0], [9, 2]], [10/11, 1/11], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
21. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[8, 0], [10, 2]], weights=[2/3, 2/3]. 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.
View answer and explanation
([[8, 0], [10, 2]], [2/3, 2/3], Yes, 4/3, No, No)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 0], [10, 2]], [2/3, 2/3], Yes, 4/3, No, No).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
22. Nonnegativity and Exact Total One as Separate Weight Tests
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], [11, 2]], weights=[22/24, 2/24]. 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.
View answer and explanation
([[9, 0], [11, 2]], [11/12, 1/12], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 0], [11, 2]], [11/12, 1/12], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
23. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[10, 0], [12, 2]], weights=[23/25, 2/25]. 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.
View answer and explanation
([[10, 0], [12, 2]], [23/25, 2/25], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 0], [12, 2]], [23/25, 2/25], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
24. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[11, 0], [13, 2]], weights=[24/26, 2/26]. 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.
View answer and explanation
([[11, 0], [13, 2]], [12/13, 1/13], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 0], [13, 2]], [12/13, 1/13], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
25. Nonnegativity and Exact Total One as Separate Weight Tests
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=[[12, 0], [14, 2]], weights=[25/27, 2/27]. 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.
View answer and explanation
([[12, 0], [14, 2]], [25/27, 2/27], Yes, 1, Yes, Yes)
Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 0], [14, 2]], [25/27, 2/27], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-qualification · candidate
26. An Exact Point from Qualified Convex Weights
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=[[-12, 0], [-10, 2]], weights=[1/3, 2/3]. 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.
View answer and explanation
([[-12, 0], [-10, 2]], [1/3, 2/3], [-32/3, 4/3])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-12, 0], [-10, 2]], [1/3, 2/3], [-32/3, 4/3]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
27. An Exact Point from Qualified Convex Weights
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=[[-11, 0], [-9, 2]], weights=[2/4, 2/4]. 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.
View answer and explanation
([[-11, 0], [-9, 2]], [1/2, 1/2], [-10, 1])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-11, 0], [-9, 2]], [1/2, 1/2], [-10, 1]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
28. An Exact Point from Qualified Convex Weights
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=[[-10, 0], [-8, 2]], weights=[3/5, 2/5]. 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.
View answer and explanation
([[-10, 0], [-8, 2]], [3/5, 2/5], [-46/5, 4/5])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-10, 0], [-8, 2]], [3/5, 2/5], [-46/5, 4/5]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
29. An Exact Point from Qualified Convex Weights
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.
View answer and explanation
([[-9, 0], [-7, 2]], [2/3, 1/3], [-25/3, 2/3])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-9, 0], [-7, 2]], [2/3, 1/3], [-25/3, 2/3]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
30. An Exact Point from Qualified Convex Weights
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=[[-8, 0], [-6, 2]], weights=[5/7, 2/7]. 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.
View answer and explanation
([[-8, 0], [-6, 2]], [5/7, 2/7], [-52/7, 4/7])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-8, 0], [-6, 2]], [5/7, 2/7], [-52/7, 4/7]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
31. An Exact Point from Qualified Convex Weights
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=[[-7, 0], [-5, 2]], weights=[6/8, 2/8]. 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.
View answer and explanation
([[-7, 0], [-5, 2]], [3/4, 1/4], [-13/2, 1/2])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-7, 0], [-5, 2]], [3/4, 1/4], [-13/2, 1/2]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
32. An Exact Point from Qualified Convex Weights
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=[[-6, 0], [-4, 2]], weights=[7/9, 2/9]. 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.
View answer and explanation
([[-6, 0], [-4, 2]], [7/9, 2/9], [-50/9, 4/9])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-6, 0], [-4, 2]], [7/9, 2/9], [-50/9, 4/9]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
33. An Exact Point from Qualified Convex Weights
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=[[-5, 0], [-3, 2]], weights=[8/10, 2/10]. 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.
View answer and explanation
([[-5, 0], [-3, 2]], [4/5, 1/5], [-23/5, 2/5])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-5, 0], [-3, 2]], [4/5, 1/5], [-23/5, 2/5]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
34. An Exact Point from Qualified Convex Weights
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=[[-4, 0], [-2, 2]], weights=[9/11, 2/11]. 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.
View answer and explanation
([[-4, 0], [-2, 2]], [9/11, 2/11], [-40/11, 4/11])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-4, 0], [-2, 2]], [9/11, 2/11], [-40/11, 4/11]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
35. An Exact Point from Qualified Convex Weights
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=[[-3, 0], [-1, 2]], weights=[10/12, 2/12]. 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.
View answer and explanation
([[-3, 0], [-1, 2]], [5/6, 1/6], [-8/3, 1/3])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-3, 0], [-1, 2]], [5/6, 1/6], [-8/3, 1/3]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
36. An Exact Point from Qualified Convex Weights
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], [0, 2]], weights=[11/13, 2/13]. 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.
View answer and explanation
([[-2, 0], [0, 2]], [11/13, 2/13], [-22/13, 4/13])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-2, 0], [0, 2]], [11/13, 2/13], [-22/13, 4/13]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
37. An Exact Point from Qualified Convex Weights
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]. 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.
View answer and explanation
([[-1, 0], [1, 2]], [6/7, 1/7], [-5/7, 2/7])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-1, 0], [1, 2]], [6/7, 1/7], [-5/7, 2/7]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
38. An Exact Point from Qualified Convex Weights
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=[[0, 0], [2, 2]], weights=[13/15, 2/15]. 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.
View answer and explanation
([[0, 0], [2, 2]], [13/15, 2/15], [4/15, 4/15])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 2]], [13/15, 2/15], [4/15, 4/15]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
39. An Exact Point from Qualified Convex Weights
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], [3, 2]], weights=[14/16, 2/16]. 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.
View answer and explanation
([[1, 0], [3, 2]], [7/8, 1/8], [5/4, 1/4])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 2]], [7/8, 1/8], [5/4, 1/4]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
40. An Exact Point from Qualified Convex Weights
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.
View answer and explanation
([[2, 0], [4, 2]], [15/17, 2/17], [38/17, 4/17])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 0], [4, 2]], [15/17, 2/17], [38/17, 4/17]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
41. An Exact Point from Qualified Convex Weights
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=[[3, 0], [5, 2]], weights=[16/18, 2/18]. 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.
View answer and explanation
([[3, 0], [5, 2]], [8/9, 1/9], [29/9, 2/9])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 0], [5, 2]], [8/9, 1/9], [29/9, 2/9]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
42. An Exact Point from Qualified Convex Weights
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=[[4, 0], [6, 2]], weights=[17/19, 2/19]. 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.
View answer and explanation
([[4, 0], [6, 2]], [17/19, 2/19], [80/19, 4/19])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 0], [6, 2]], [17/19, 2/19], [80/19, 4/19]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
43. An Exact Point from Qualified Convex Weights
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=[[5, 0], [7, 2]], weights=[18/20, 2/20]. 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.
View answer and explanation
([[5, 0], [7, 2]], [9/10, 1/10], [26/5, 1/5])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 0], [7, 2]], [9/10, 1/10], [26/5, 1/5]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
44. An Exact Point from Qualified Convex Weights
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=[[6, 0], [8, 2]], weights=[19/21, 2/21]. 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.
View answer and explanation
([[6, 0], [8, 2]], [19/21, 2/21], [130/21, 4/21])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 0], [8, 2]], [19/21, 2/21], [130/21, 4/21]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
45. An Exact Point from Qualified Convex Weights
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=[[7, 0], [9, 2]], weights=[20/22, 2/22]. 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.
View answer and explanation
([[7, 0], [9, 2]], [10/11, 1/11], [79/11, 2/11])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 0], [9, 2]], [10/11, 1/11], [79/11, 2/11]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
46. An Exact Point from Qualified Convex Weights
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=[[8, 0], [10, 2]], weights=[21/23, 2/23]. 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.
View answer and explanation
([[8, 0], [10, 2]], [21/23, 2/23], [188/23, 4/23])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 0], [10, 2]], [21/23, 2/23], [188/23, 4/23]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
47. An Exact Point from Qualified Convex Weights
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], [11, 2]], weights=[22/24, 2/24]. 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.
View answer and explanation
([[9, 0], [11, 2]], [11/12, 1/12], [55/6, 1/6])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 0], [11, 2]], [11/12, 1/12], [55/6, 1/6]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
48. An Exact Point from Qualified Convex Weights
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=[[10, 0], [12, 2]], weights=[23/25, 2/25]. 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.
View answer and explanation
([[10, 0], [12, 2]], [23/25, 2/25], [254/25, 4/25])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 0], [12, 2]], [23/25, 2/25], [254/25, 4/25]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
49. An Exact Point from Qualified Convex Weights
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=[[11, 0], [13, 2]], weights=[24/26, 2/26]. 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.
View answer and explanation
([[11, 0], [13, 2]], [12/13, 1/13], [145/13, 2/13])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 0], [13, 2]], [12/13, 1/13], [145/13, 2/13]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
50. An Exact Point from Qualified Convex Weights
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=[[12, 0], [14, 2]], weights=[25/27, 2/27]. 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.
View answer and explanation
([[12, 0], [14, 2]], [25/27, 2/27], [328/27, 4/27])
Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 0], [14, 2]], [25/27, 2/27], [328/27, 4/27]).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-weighted-point · candidate
51. Audit lambda without Clipping an Affine Extension
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=[-9, 3], lambda=1/3. 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.
View answer and explanation
([-12, 0], [-9, 3], 1/3, Yes, [-10, 2], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 0], [-9, 3], 1/3, Yes, [-10, 2], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
52. Audit lambda without Clipping an Affine Extension
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=[-11, 0], right=[-8, 3], lambda=2/4. 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.
View answer and explanation
([-11, 0], [-8, 3], 1/2, Yes, [-19/2, 3/2], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 0], [-8, 3], 1/2, Yes, [-19/2, 3/2], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
53. Audit lambda without Clipping an Affine Extension
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=[-10, 0], right=[-7, 3], lambda=3/5. 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.
View answer and explanation
([-10, 0], [-7, 3], 3/5, Yes, [-44/5, 6/5], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0], [-7, 3], 3/5, Yes, [-44/5, 6/5], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
54. Audit lambda without Clipping an Affine Extension
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=[-9, 0], right=[-6, 3], lambda=4/6. 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.
View answer and explanation
([-9, 0], [-6, 3], 2/3, Yes, [-8, 1], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-6, 3], 2/3, Yes, [-8, 1], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
55. Audit lambda without Clipping an Affine Extension
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=[-8, 0], right=[-5, 3], lambda=5/7. 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.
View answer and explanation
([-8, 0], [-5, 3], 5/7, Yes, [-50/7, 6/7], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 0], [-5, 3], 5/7, Yes, [-50/7, 6/7], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
56. Audit lambda without Clipping an Affine Extension
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=[-7, 0], right=[-4, 3], lambda=6/8. 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.
View answer and explanation
([-7, 0], [-4, 3], 3/4, Yes, [-25/4, 3/4], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 0], [-4, 3], 3/4, Yes, [-25/4, 3/4], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
57. Audit lambda without Clipping an Affine Extension
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=[-6, 0], right=[-3, 3], lambda=7/9. 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.
View answer and explanation
([-6, 0], [-3, 3], 7/9, Yes, [-16/3, 2/3], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 0], [-3, 3], 7/9, Yes, [-16/3, 2/3], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
58. Audit lambda without Clipping an Affine Extension
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=[-2, 3], lambda=8/10. 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.
View answer and explanation
([-5, 0], [-2, 3], 4/5, Yes, [-22/5, 3/5], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0], [-2, 3], 4/5, Yes, [-22/5, 3/5], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
59. Audit lambda without Clipping an Affine Extension
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=[-4, 0], right=[-1, 3], lambda=9/11. 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.
View answer and explanation
([-4, 0], [-1, 3], 9/11, Yes, [-38/11, 6/11], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 0], [-1, 3], 9/11, Yes, [-38/11, 6/11], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
60. Audit lambda without Clipping an Affine Extension
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=[-3, 0], right=[0, 3], lambda=10/12. 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.
View answer and explanation
([-3, 0], [0, 3], 5/6, Yes, [-5/2, 1/2], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 0], [0, 3], 5/6, Yes, [-5/2, 1/2], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
61. Audit lambda without Clipping an Affine Extension
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=[-2, 0], right=[1, 3], lambda=11/13. 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.
View answer and explanation
([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
62. Audit lambda without Clipping an Affine Extension
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=[-1, 0], right=[2, 3], lambda=12/14. 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.
View answer and explanation
([-1, 0], [2, 3], 6/7, Yes, [-4/7, 3/7], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 0], [2, 3], 6/7, Yes, [-4/7, 3/7], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
63. Audit lambda without Clipping an Affine Extension
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=[0, 0], right=[3, 3], lambda=13/15. 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.
View answer and explanation
([0, 0], [3, 3], 13/15, Yes, [2/5, 2/5], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0], [3, 3], 13/15, Yes, [2/5, 2/5], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
64. Audit lambda without Clipping an Affine Extension
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=[1, 0], right=[4, 3], lambda=14/16. 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.
View answer and explanation
([1, 0], [4, 3], 7/8, Yes, [11/8, 3/8], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0], [4, 3], 7/8, Yes, [11/8, 3/8], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
65. Audit lambda without Clipping an Affine Extension
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=[2, 0], right=[5, 3], lambda=15/17. 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.
View answer and explanation
([2, 0], [5, 3], 15/17, Yes, [40/17, 6/17], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0], [5, 3], 15/17, Yes, [40/17, 6/17], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
66. Audit lambda without Clipping an Affine Extension
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=[3, 0], right=[6, 3], lambda=16/18. 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.
View answer and explanation
([3, 0], [6, 3], 8/9, Yes, [10/3, 1/3], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], [6, 3], 8/9, Yes, [10/3, 1/3], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
67. Audit lambda without Clipping an Affine Extension
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=[4, 0], right=[7, 3], lambda=17/19. 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.
View answer and explanation
([4, 0], [7, 3], 17/19, Yes, [82/19, 6/19], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, 0], [7, 3], 17/19, Yes, [82/19, 6/19], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
68. Audit lambda without Clipping an Affine Extension
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.
View answer and explanation
([5, 0], [8, 3], 9/10, Yes, [53/10, 3/10], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [8, 3], 9/10, Yes, [53/10, 3/10], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
69. Audit lambda without Clipping an Affine Extension
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=[6, 0], right=[9, 3], lambda=19/21. 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.
View answer and explanation
([6, 0], [9, 3], 19/21, Yes, [44/7, 2/7], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 0], [9, 3], 19/21, Yes, [44/7, 2/7], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
70. Audit lambda without Clipping an Affine Extension
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=[7, 0], right=[10, 3], lambda=20/22. 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.
View answer and explanation
([7, 0], [10, 3], 10/11, Yes, [80/11, 3/11], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 0], [10, 3], 10/11, Yes, [80/11, 3/11], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
71. Audit lambda without Clipping an Affine Extension
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=[8, 0], right=[11, 3], lambda=21/23. 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.
View answer and explanation
([8, 0], [11, 3], 21/23, Yes, [190/23, 6/23], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, 0], [11, 3], 21/23, Yes, [190/23, 6/23], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
72. Audit lambda without Clipping an Affine Extension
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=[9, 0], right=[12, 3], lambda=22/24. 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.
View answer and explanation
([9, 0], [12, 3], 11/12, Yes, [37/4, 1/4], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, 0], [12, 3], 11/12, Yes, [37/4, 1/4], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
73. Audit lambda without Clipping an Affine Extension
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=[10, 0], right=[13, 3], lambda=23/25. 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.
View answer and explanation
([10, 0], [13, 3], 23/25, Yes, [256/25, 6/25], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0], [13, 3], 23/25, Yes, [256/25, 6/25], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
74. Audit lambda without Clipping an Affine Extension
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=[11, 0], right=[14, 3], lambda=24/26. 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.
View answer and explanation
([11, 0], [14, 3], 12/13, Yes, [146/13, 3/13], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 0], [14, 3], 12/13, Yes, [146/13, 3/13], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
75. Audit lambda without Clipping an Affine Extension
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.
View answer and explanation
([12, 0], [15, 3], 25/27, Yes, [110/9, 2/9], Yes)
Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [15, 3], 25/27, Yes, [110/9, 2/9], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-two-point · candidate
76. A Qualified Convex Witness versus Its Target
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=[[0, 0], [3, 0]], weights=[1/3, 2/3], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [3, 0]], [1/3, 2/3], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [3, 0]], [1/3, 2/3], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
77. A Qualified Convex Witness versus Its Target
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=[[0, 0], [4, 0]], weights=[2/4, 2/4], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [4, 0]], [1/2, 1/2], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [4, 0]], [1/2, 1/2], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
78. A Qualified Convex Witness versus Its Target
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=[[0, 0], [5, 0]], weights=[3/5, 2/5], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [5, 0]], [3/5, 2/5], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [5, 0]], [3/5, 2/5], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
79. A Qualified Convex Witness versus Its Target
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=[[0, 0], [6, 0]], weights=[4/6, 2/6], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [6, 0]], [2/3, 1/3], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [6, 0]], [2/3, 1/3], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
80. A Qualified Convex Witness versus Its Target
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=[[0, 0], [7, 0]], weights=[5/7, 2/7], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [7, 0]], [5/7, 2/7], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [7, 0]], [5/7, 2/7], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
81. A Qualified Convex Witness versus Its Target
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=[[0, 0], [8, 0]], weights=[6/8, 2/8], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [8, 0]], [3/4, 1/4], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [8, 0]], [3/4, 1/4], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
82. A Qualified Convex Witness versus Its Target
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=[[0, 0], [9, 0]], weights=[7/9, 2/9], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [9, 0]], [7/9, 2/9], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [9, 0]], [7/9, 2/9], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
83. A Qualified Convex Witness versus Its Target
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=[[0, 0], [10, 0]], weights=[8/10, 2/10], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [10, 0]], [4/5, 1/5], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [10, 0]], [4/5, 1/5], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
84. A Qualified Convex Witness versus Its Target
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=[[0, 0], [11, 0]], weights=[9/11, 2/11], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [11, 0]], [9/11, 2/11], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [11, 0]], [9/11, 2/11], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
85. A Qualified Convex Witness versus Its Target
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=[[0, 0], [12, 0]], weights=[10/12, 2/12], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [12, 0]], [5/6, 1/6], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [12, 0]], [5/6, 1/6], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
86. A Qualified Convex Witness versus Its Target
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=[[0, 0], [13, 0]], weights=[11/13, 2/13], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [13, 0]], [11/13, 2/13], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [13, 0]], [11/13, 2/13], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
87. A Qualified Convex Witness versus Its Target
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=[[0, 0], [14, 0]], weights=[12/14, 2/14], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [14, 0]], [6/7, 1/7], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [14, 0]], [6/7, 1/7], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
88. A Qualified Convex Witness versus Its Target
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=[[0, 0], [15, 0]], weights=[13/15, 2/15], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [15, 0]], [13/15, 2/15], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [15, 0]], [13/15, 2/15], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
89. A Qualified Convex Witness versus Its Target
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=[[0, 0], [16, 0]], weights=[14/16, 2/16], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [16, 0]], [7/8, 1/8], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [16, 0]], [7/8, 1/8], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
90. A Qualified Convex Witness versus Its Target
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=[[0, 0], [17, 0]], weights=[15/17, 2/17], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [17, 0]], [15/17, 2/17], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [17, 0]], [15/17, 2/17], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
91. A Qualified Convex Witness versus Its Target
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=[[0, 0], [18, 0]], weights=[16/18, 2/18], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [18, 0]], [8/9, 1/9], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [18, 0]], [8/9, 1/9], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
92. A Qualified Convex Witness versus Its Target
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=[[0, 0], [19, 0]], weights=[17/19, 2/19], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [19, 0]], [17/19, 2/19], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [19, 0]], [17/19, 2/19], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
93. A Qualified Convex Witness versus Its Target
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=[[0, 0], [20, 0]], weights=[18/20, 2/20], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [20, 0]], [9/10, 1/10], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [20, 0]], [9/10, 1/10], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
94. A Qualified Convex Witness versus Its Target
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=[[0, 0], [21, 0]], weights=[19/21, 2/21], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [21, 0]], [19/21, 2/21], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [21, 0]], [19/21, 2/21], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
95. A Qualified Convex Witness versus Its Target
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=[[0, 0], [22, 0]], weights=[20/22, 2/22], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [22, 0]], [10/11, 1/11], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [22, 0]], [10/11, 1/11], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
96. A Qualified Convex Witness versus Its Target
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=[[0, 0], [23, 0]], weights=[21/23, 2/23], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [23, 0]], [21/23, 2/23], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [23, 0]], [21/23, 2/23], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
97. A Qualified Convex Witness versus Its Target
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=[[0, 0], [24, 0]], weights=[22/24, 2/24], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [24, 0]], [11/12, 1/12], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [24, 0]], [11/12, 1/12], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
98. A Qualified Convex Witness versus Its Target
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=[[0, 0], [25, 0]], weights=[23/25, 2/25], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [25, 0]], [23/25, 2/25], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [25, 0]], [23/25, 2/25], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
99. A Qualified Convex Witness versus Its Target
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=[[0, 0], [26, 0]], weights=[24/26, 2/26], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [26, 0]], [12/13, 1/13], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [26, 0]], [12/13, 1/13], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
100. A Qualified Convex Witness versus Its Target
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=[[0, 0], [27, 0]], weights=[25/27, 2/27], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[0, 0], [27, 0]], [25/27, 2/27], [2, 0], [2, 0], Yes)
Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [27, 0]], [25/27, 2/27], [2, 0], [2, 0], Yes).
Reviewed step-by-step derivation
Finite convex combination · qf-mod24-convcomb-witness-audit · candidate
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