Explicit Span, Dependence and Ordered-Basis Coordinate Witnesses
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batch-20261013-advance-20261025-r04-linear-independence-basis-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. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-12, -3], target=[-11, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-12, -3], [-11, -3], [-12, -3], No)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-12, -3], [-11, -3], [-12, -3], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
2. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-11, -2], target=[-11, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
3. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-10, -1], target=[-10, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
4. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-9, 0], target=[-9, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
5. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-8, 1], target=[-8, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
6. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-7, 2], target=[-6, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], No)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
7. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-6, 3], target=[-6, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
8. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-5, -3], target=[-5, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
9. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-4, -2], target=[-4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
10. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-3, -1], target=[-3, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
11. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-2, 0], target=[-1, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], No)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
12. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-1, 1], target=[-1, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
13. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[0, 2], target=[0, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
14. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[1, 3], target=[1, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
15. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[2, -3], target=[2, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
16. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[3, -2], target=[4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], No)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
17. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[4, -1], target=[4, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
18. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[5, 0], target=[5, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
19. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[6, 1], target=[6, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
20. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[7, 2], target=[7, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
21. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[8, 3], target=[9, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], No)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
22. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[9, -3], target=[9, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
23. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[10, -2], target=[10, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
24. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[11, -1], target=[11, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
25. Explicit Reconstruction without Conflating Span and Independence
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[12, 0], target=[12, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).
View answer and explanation
([[1, 0], [0, 1]], [12, 0], [12, 0], [12, 0], Yes)
A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [12, 0], [12, 0], [12, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-span-witness-audit · candidate
26. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [-2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 0], [-2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
27. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -11], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -11], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -11], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
28. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -10], [0, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -10], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -10], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
29. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[0, 0], [1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
30. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -8], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -8], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -8], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
31. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -7], [-2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
32. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [-1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
33. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -5], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -5], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -5], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
34. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -4], [1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -4], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -4], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
35. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[0, 0], [2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
36. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -2], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -2], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
37. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -1], [-1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
38. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 0], [0, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
39. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 1], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 1], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
40. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 2], [2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 2], [2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
41. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [-2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
42. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 4], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 4], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 4], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
43. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 5], [0, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 5], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 5], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
44. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 0], [1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
45. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 7], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 7], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 7], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
46. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 8], [-2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
47. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [-1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
48. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 10], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 10], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 10], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
49. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 11], [1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 11], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 11], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
50. A Selected Zero Vector Gives a Nontrivial Zero Relation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.
View answer and explanation
([[1, 0], [2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)
The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-zero-vector-dependence-certificate · candidate
51. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[1, 2], right=[1, 3], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([1, 2], [1, 3], 1, [1, 2], [0, -1], No)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], [1, 3], 1, [1, 2], [0, -1], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
52. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[2, 3], right=[4, 6], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([2, 3], [4, 6], 2, [4, 6], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], [4, 6], 2, [4, 6], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
53. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[3, 4], right=[9, 12], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([3, 4], [9, 12], 3, [9, 12], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 4], [9, 12], 3, [9, 12], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
54. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[4, 5], right=[16, 20], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([4, 5], [16, 20], 4, [16, 20], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, 5], [16, 20], 4, [16, 20], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
55. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[5, 6], right=[25, 30], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([5, 6], [25, 30], 5, [25, 30], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 6], [25, 30], 5, [25, 30], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
56. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[6, 7], right=[6, 7], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([6, 7], [6, 7], 1, [6, 7], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 7], [6, 7], 1, [6, 7], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
57. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[7, 8], right=[14, 17], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([7, 8], [14, 17], 2, [14, 16], [0, -1], No)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 8], [14, 17], 2, [14, 16], [0, -1], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
58. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[8, 9], right=[24, 27], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([8, 9], [24, 27], 3, [24, 27], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, 9], [24, 27], 3, [24, 27], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
59. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[9, 10], right=[36, 40], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([9, 10], [36, 40], 4, [36, 40], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, 10], [36, 40], 4, [36, 40], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
60. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[10, 11], right=[50, 55], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([10, 11], [50, 55], 5, [50, 55], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 11], [50, 55], 5, [50, 55], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
61. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[11, 12], right=[11, 12], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([11, 12], [11, 12], 1, [11, 12], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 12], [11, 12], 1, [11, 12], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
62. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[12, 13], right=[24, 26], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([12, 13], [24, 26], 2, [24, 26], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 13], [24, 26], 2, [24, 26], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
63. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[13, 14], right=[39, 43], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([13, 14], [39, 43], 3, [39, 42], [0, -1], No)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([13, 14], [39, 43], 3, [39, 42], [0, -1], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
64. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[14, 15], right=[56, 60], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([14, 15], [56, 60], 4, [56, 60], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([14, 15], [56, 60], 4, [56, 60], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
65. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[15, 16], right=[75, 80], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([15, 16], [75, 80], 5, [75, 80], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([15, 16], [75, 80], 5, [75, 80], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
66. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[16, 17], right=[16, 17], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([16, 17], [16, 17], 1, [16, 17], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([16, 17], [16, 17], 1, [16, 17], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
67. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[17, 18], right=[34, 36], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([17, 18], [34, 36], 2, [34, 36], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([17, 18], [34, 36], 2, [34, 36], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
68. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[18, 19], right=[54, 57], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([18, 19], [54, 57], 3, [54, 57], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([18, 19], [54, 57], 3, [54, 57], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
69. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[19, 20], right=[76, 81], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([19, 20], [76, 81], 4, [76, 80], [0, -1], No)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([19, 20], [76, 81], 4, [76, 80], [0, -1], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
70. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[20, 21], right=[100, 105], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([20, 21], [100, 105], 5, [100, 105], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([20, 21], [100, 105], 5, [100, 105], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
71. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[21, 22], right=[21, 22], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([21, 22], [21, 22], 1, [21, 22], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([21, 22], [21, 22], 1, [21, 22], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
72. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[22, 23], right=[44, 46], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([22, 23], [44, 46], 2, [44, 46], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([22, 23], [44, 46], 2, [44, 46], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
73. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[23, 24], right=[69, 72], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([23, 24], [69, 72], 3, [69, 72], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([23, 24], [69, 72], 3, [69, 72], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
74. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[24, 25], right=[96, 100], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([24, 25], [96, 100], 4, [96, 100], [0, 0], Yes)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([24, 25], [96, 100], 4, [96, 100], [0, 0], Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
75. A Nonzero Proportional-Pair Witness Must Actually Match
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[25, 26], right=[125, 131], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).
View answer and explanation
([25, 26], [125, 131], 5, [125, 130], [0, -1], No)
Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([25, 26], [125, 131], 5, [125, 130], [0, -1], No).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-proportional-pair-dependence-certificate · candidate
76. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-12, 21], first integer coordinates=[-12, -3], second=[-11, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], Yes, [-11, -3], [-11, 19], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], Yes, [-11, -3], [-11, 19], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
77. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-11, 9], first integer coordinates=[-11, -2], second=[-11, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], Yes, [-11, -2], [-11, 9], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], Yes, [-11, -2], [-11, 9], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
78. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[-10, -1], first integer coordinates=[-10, -1], second=[-10, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], Yes, [-10, -1], [-10, -1], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], Yes, [-10, -1], [-10, -1], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
79. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[-9, -9], first integer coordinates=[-9, 0], second=[-9, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], Yes, [-9, 0], [-9, -9], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], Yes, [-9, 0], [-9, -9], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
80. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[-8, -15], first integer coordinates=[-8, 1], second=[-7, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], Yes, [-7, 1], [-7, -13], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], Yes, [-7, 1], [-7, -13], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
81. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-7, 16], first integer coordinates=[-7, 2], second=[-7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], Yes, [-7, 2], [-7, 16], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], Yes, [-7, 2], [-7, 16], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
82. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-6, 9], first integer coordinates=[-6, 3], second=[-6, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], Yes, [-6, 3], [-6, 9], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], Yes, [-6, 3], [-6, 9], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
83. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[-5, -3], first integer coordinates=[-5, -3], second=[-5, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], Yes, [-5, -3], [-5, -3], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], Yes, [-5, -3], [-5, -3], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
84. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[-4, -6], first integer coordinates=[-4, -2], second=[-3, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], Yes, [-3, -2], [-3, -5], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], Yes, [-3, -2], [-3, -5], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
85. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[-3, -7], first integer coordinates=[-3, -1], second=[-3, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], Yes, [-3, -1], [-3, -7], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], Yes, [-3, -1], [-3, -7], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
86. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-2, 4], first integer coordinates=[-2, 0], second=[-2, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
87. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-1, 2], first integer coordinates=[-1, 1], second=[-1, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], Yes, [-1, 1], [-1, 2], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], Yes, [-1, 1], [-1, 2], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
88. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[0, 2], first integer coordinates=[0, 2], second=[1, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], Yes, [1, 2], [1, 2], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], Yes, [1, 2], [1, 2], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
89. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[1, 4], first integer coordinates=[1, 3], second=[1, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], Yes, [1, 3], [1, 4], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], Yes, [1, 3], [1, 4], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
90. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[2, 1], first integer coordinates=[2, -3], second=[2, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], Yes, [2, -3], [2, 1], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], Yes, [2, -3], [2, 1], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
91. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[3, -8], first integer coordinates=[3, -2], second=[3, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], Yes, [3, -2], [3, -8], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], Yes, [3, -2], [3, -8], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
92. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[4, -5], first integer coordinates=[4, -1], second=[5, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], Yes, [5, -1], [5, -6], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], Yes, [5, -1], [5, -6], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
93. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[5, 0], first integer coordinates=[5, 0], second=[5, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], Yes, [5, 0], [5, 0], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], Yes, [5, 0], [5, 0], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
94. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[6, 7], first integer coordinates=[6, 1], second=[6, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
95. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[7, 16], first integer coordinates=[7, 2], second=[7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
96. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[8, -13], first integer coordinates=[8, 3], second=[9, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], Yes, [9, 3], [9, -15], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], Yes, [9, 3], [9, -15], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
97. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[9, -12], first integer coordinates=[9, -3], second=[9, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], Yes, [9, -3], [9, -12], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], Yes, [9, -3], [9, -12], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
98. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[10, -2], first integer coordinates=[10, -2], second=[10, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], Yes, [10, -2], [10, -2], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], Yes, [10, -2], [10, -2], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
99. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[11, 10], first integer coordinates=[11, -1], second=[11, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], Yes, [11, -1], [11, 10], Yes, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], Yes, [11, -1], [11, 10], Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
100. Candidate Coordinate Lists and the Conditional Uniqueness Obligation
Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[12, 24], first integer coordinates=[12, 0], second=[13, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).
View answer and explanation
([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], Yes, [13, 0], [13, 26], No, Yes)
Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], Yes, [13, 0], [13, 26], No, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod25-basisdeep-r2-coordinate-uniqueness-audit · candidate
Canonical question JSONL · English JSONL · Validation report