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Finite Real-Coordinate Linear Combinations and Span Witnesses

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batch-20261013-advance-20261023-r02-linear-combination-span-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. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-12, -3, -2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

2. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-11, -2, -1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

3. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-10, -1, 0]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

4. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-9, 0, 1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

5. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-8, 1, 2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

6. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-7, 2, -2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

7. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-6, 3, -1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

8. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

9. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

10. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-3, -1, 2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

11. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-2, 0, -2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

12. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, 1, -1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

13. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[0, 2, 0]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

14. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[1, 3, 1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

15. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[2, -3, 2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

16. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[3, -2, -2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

17. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[4, -1, -1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

18. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[5, 0, 0]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

19. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[6, 1, 1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

20. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[7, 2, 2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

21. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[8, 3, -2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

22. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[9, -3, -1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

23. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[10, -2, 0]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

24. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

25. An Ordered Generator-Coefficient Weighted Sum

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[12, 0, 2]. Give (original vectors,original coefficients,the coordinate sum).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2])

Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-weighted-sum · candidate

26. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-12, -3, -2], target=[-13, -5]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

27. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-11, -2, -1], target=[-12, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

28. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-10, -1, 0], target=[-10, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

29. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-9, 0, 1], target=[-8, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

30. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-8, 1, 2], target=[-5, 3]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

31. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-7, 2, -2], target=[-9, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

32. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-6, 3, -1], target=[-7, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

33. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0], target=[-5, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

34. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1], target=[-2, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

35. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-3, -1, 2], target=[-1, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

36. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-2, 0, -2], target=[-4, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

37. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, 1, -1], target=[-2, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

38. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[0, 2, 0], target=[1, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

39. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[1, 3, 1], target=[2, 4]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

40. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[2, -3, 2], target=[4, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

41. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[3, -2, -2], target=[1, -4]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

42. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[4, -1, -1], target=[4, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

43. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[5, 0, 0], target=[5, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

44. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[6, 1, 1], target=[7, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

45. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[7, 2, 2], target=[9, 4]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

46. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[8, 3, -2], target=[7, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

47. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[9, -3, -1], target=[8, -4]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

48. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[10, -2, 0], target=[10, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

49. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1], target=[12, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], Yes)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

50. A Given Coefficient Witness against a Target Vector

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[12, 0, 2], target=[15, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).

View answer and explanation

([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], No)

Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-witness-audit · candidate

51. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-12, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

52. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-11, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

53. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-10, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

54. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-9, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

55. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-8, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

56. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

57. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-6, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

58. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-5, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

59. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-4, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

60. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-3, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

61. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-2, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

62. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-1, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

63. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[0, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

64. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[1, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

65. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[2, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

66. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[3, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

67. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[4, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

68. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[5, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

69. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[6, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

70. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

71. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[8, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

72. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[9, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

73. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[10, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

74. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[11, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

75. Unique Real Coordinates in Two Independent Plane Generators

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[12, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0])

Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0]).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-r2-coordinate-solve · candidate

76. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-20, -39]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-20, -39], -10, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-20, -39], -10, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

77. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-18, -36]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-18, -36], -9, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-18, -36], -9, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

78. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-16, -32]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-16, -32], -8, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-16, -32], -8, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

79. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-14, -28]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-14, -28], -7, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-14, -28], -7, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

80. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-12, -23]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-12, -23], -6, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-12, -23], -6, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

81. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-10, -20]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-10, -20], -5, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-10, -20], -5, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

82. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-8, -16]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-8, -16], -4, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-8, -16], -4, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

83. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-6, -12]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-6, -12], -3, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-6, -12], -3, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

84. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-4, -7]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-4, -7], -2, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-4, -7], -2, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

85. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-2, -4]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [-2, -4], -1, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-2, -4], -1, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

86. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[0, 0]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [0, 0], 0, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [0, 0], 0, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

87. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[2, 4]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [2, 4], 1, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [2, 4], 1, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

88. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[4, 9]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [4, 9], 2, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [4, 9], 2, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

89. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[6, 12]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [6, 12], 3, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [6, 12], 3, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

90. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[8, 16]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [8, 16], 4, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [8, 16], 4, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

91. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[10, 20]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [10, 20], 5, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [10, 20], 5, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

92. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[12, 25]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [12, 25], 6, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [12, 25], 6, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

93. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[14, 28]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [14, 28], 7, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [14, 28], 7, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

94. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[16, 32]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [16, 32], 8, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [16, 32], 8, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

95. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[18, 36]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [18, 36], 9, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [18, 36], 9, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

96. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[20, 41]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [20, 41], 10, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [20, 41], 10, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

97. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[22, 44]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [22, 44], 11, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [22, 44], 11, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

98. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[24, 48]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [24, 48], 12, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [24, 48], 12, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

99. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[26, 52]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [26, 52], 13, Yes)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [26, 52], 13, Yes).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

100. A Single Generator, Including the Zero-Vector Case

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[28, 57]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

View answer and explanation

([2, 4], [28, 57], 14, No)

Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [28, 57], 14, No).

Reviewed step-by-step derivation

Linear combinations and span · qf-mod23-span-single-generator-membership · candidate

Canonical question JSONL · English JSONL · Validation report