Exact Induced Norms, Distances, Cauchy–Schwarz and Parallelogram Checks
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batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[2, 0], [0, 1]] and x=[-3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[2, 0], [0, 1]], [-3, -2], 22, sqrt(22), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[2, 0], [0, 1]], [-3, -2], 22, sqrt(22), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
2. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[3, 1], [1, 2]] and x=[-2, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[3, 1], [1, 2]], [-2, -1], 18, sqrt(18), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[3, 1], [1, 2]], [-2, -1], 18, sqrt(18), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
3. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[4, 2], [2, 5]] and x=[-1, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[4, 2], [2, 5]], [-1, 0], 4, sqrt(4), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[4, 2], [2, 5]], [-1, 0], 4, sqrt(4), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
4. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[5, 0], [0, 1]] and x=[0, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[5, 0], [0, 1]], [0, 1], 1, sqrt(1), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[5, 0], [0, 1]], [0, 1], 1, sqrt(1), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
5. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[6, 1], [1, 2]] and x=[1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[6, 1], [1, 2]], [1, 2], 18, sqrt(18), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[6, 1], [1, 2]], [1, 2], 18, sqrt(18), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
6. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[7, 2], [2, 5]] and x=[2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[7, 2], [2, 5]], [2, -2], 32, sqrt(32), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[7, 2], [2, 5]], [2, -2], 32, sqrt(32), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
7. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
8. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[9, 1], [1, 2]] and x=[-3, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[9, 1], [1, 2]], [-3, 0], 81, sqrt(81), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[9, 1], [1, 2]], [-3, 0], 81, sqrt(81), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
9. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[10, 2], [2, 5]], [-2, 1], 37, sqrt(37), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[10, 2], [2, 5]], [-2, 1], 37, sqrt(37), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
10. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[11, 0], [0, 1]], [-1, 2], 15, sqrt(15), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [-1, 2], 15, sqrt(15), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
11. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[12, 1], [1, 2]] and x=[0, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[12, 1], [1, 2]], [0, -2], 8, sqrt(8), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], [0, -2], 8, sqrt(8), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
12. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[13, 2], [2, 5]] and x=[1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[13, 2], [2, 5]], [1, -1], 14, sqrt(14), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [1, -1], 14, sqrt(14), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
13. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[14, 0], [0, 1]] and x=[2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[14, 0], [0, 1]], [2, 0], 56, sqrt(56), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[14, 0], [0, 1]], [2, 0], 56, sqrt(56), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
14. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[15, 1], [1, 2]] and x=[3, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[15, 1], [1, 2]], [3, 1], 143, sqrt(143), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[15, 1], [1, 2]], [3, 1], 143, sqrt(143), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
15. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[16, 2], [2, 5]] and x=[-3, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[16, 2], [2, 5]], [-3, 2], 140, sqrt(140), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[16, 2], [2, 5]], [-3, 2], 140, sqrt(140), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
16. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[17, 0], [0, 1]] and x=[-2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[17, 0], [0, 1]], [-2, -2], 72, sqrt(72), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[17, 0], [0, 1]], [-2, -2], 72, sqrt(72), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
17. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[18, 1], [1, 2]] and x=[-1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[18, 1], [1, 2]], [-1, -1], 22, sqrt(22), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[18, 1], [1, 2]], [-1, -1], 22, sqrt(22), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
18. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[19, 2], [2, 5]], [0, 0], 0, sqrt(0), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[19, 2], [2, 5]], [0, 0], 0, sqrt(0), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
19. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[20, 0], [0, 1]] and x=[1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[20, 0], [0, 1]], [1, 1], 21, sqrt(21), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[20, 0], [0, 1]], [1, 1], 21, sqrt(21), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
20. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[21, 1], [1, 2]] and x=[2, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[21, 1], [1, 2]], [2, 2], 100, sqrt(100), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[21, 1], [1, 2]], [2, 2], 100, sqrt(100), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
21. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[22, 2], [2, 5]] and x=[3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[22, 2], [2, 5]], [3, -2], 194, sqrt(194), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[22, 2], [2, 5]], [3, -2], 194, sqrt(194), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
22. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[23, 0], [0, 1]] and x=[-3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[23, 0], [0, 1]], [-3, -1], 208, sqrt(208), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [-3, -1], 208, sqrt(208), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
23. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[24, 1], [1, 2]] and x=[-2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[24, 1], [1, 2]], [-2, 0], 96, sqrt(96), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [-2, 0], 96, sqrt(96), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
24. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[25, 2], [2, 5]] and x=[-1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[25, 2], [2, 5]], [-1, 1], 26, sqrt(26), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [-1, 1], 26, sqrt(26), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
25. Use the Nonnegative Exact Root of the Same Self-Inner-Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[26, 0], [0, 1]] and x=[0, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).
View answer and explanation
([[26, 0], [0, 1]], [0, 2], 4, sqrt(4), same_inner_product_universal_norm_certificate)
Compute x^T A x exactly, then retain sqrt(norm_square). Norm qualification uses the same certified inner product and universal Cauchy–Schwarz triangle argument, not just this sample. The exact result, retaining the requested order, is ([[26, 0], [0, 1]], [0, 2], 4, sqrt(4), same_inner_product_universal_norm_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-square · candidate
26. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[2, 0], [0, 1]] and x=[-3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 12]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-3, -2], [-5, 12], [2, -14], 204, sqrt(204))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-3, -2], [-5, 12], [2, -14], 204, sqrt(204)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
27. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[3, 1], [1, 2]] and x=[-2, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 11]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-2, -1], [-4, 11], [2, -12], 252, sqrt(252))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-2, -1], [-4, 11], [2, -12], 252, sqrt(252)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
28. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[4, 2], [2, 5]] and x=[-1, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, 10]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-1, 0], [-3, 10], [2, -10], 436, sqrt(436))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-1, 0], [-3, 10], [2, -10], 436, sqrt(436)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
29. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[5, 0], [0, 1]] and x=[0, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, 9]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([0, 1], [-2, 9], [2, -8], 84, sqrt(84))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([0, 1], [-2, 9], [2, -8], 84, sqrt(84)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
30. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[6, 1], [1, 2]] and x=[1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, 8]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([1, 2], [-1, 8], [2, -6], 72, sqrt(72))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([1, 2], [-1, 8], [2, -6], 72, sqrt(72)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
31. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[7, 2], [2, 5]] and x=[2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, 7]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([2, -2], [0, 7], [2, -9], 361, sqrt(361))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([2, -2], [0, 7], [2, -9], 361, sqrt(361)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
32. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, 6]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([3, -1], [1, 6], [2, -7], 81, sqrt(81))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([3, -1], [1, 6], [2, -7], 81, sqrt(81)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
33. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[9, 1], [1, 2]] and x=[-3, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, 5]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-3, 0], [2, 5], [-5, -5], 325, sqrt(325))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-3, 0], [2, 5], [-5, -5], 325, sqrt(325)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
34. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, 4]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-2, 1], [3, 4], [-5, -3], 355, sqrt(355))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-2, 1], [3, 4], [-5, -3], 355, sqrt(355)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
35. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, 3]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-1, 2], [4, 3], [-5, -1], 276, sqrt(276))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-1, 2], [4, 3], [-5, -1], 276, sqrt(276)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
36. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[12, 1], [1, 2]] and x=[0, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, 2]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([0, -2], [5, 2], [-5, -4], 372, sqrt(372))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([0, -2], [5, 2], [-5, -4], 372, sqrt(372)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
37. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[13, 2], [2, 5]] and x=[1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 1]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([1, -1], [-5, 1], [6, -2], 440, sqrt(440))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([1, -1], [-5, 1], [6, -2], 440, sqrt(440)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
38. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[14, 0], [0, 1]] and x=[2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 0]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([2, 0], [-4, 0], [6, 0], 504, sqrt(504))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([2, 0], [-4, 0], [6, 0], 504, sqrt(504)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
39. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[15, 1], [1, 2]] and x=[3, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -1]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([3, 1], [-3, -1], [6, 2], 572, sqrt(572))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([3, 1], [-3, -1], [6, 2], 572, sqrt(572)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
40. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[16, 2], [2, 5]] and x=[-3, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, -2]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-3, 2], [-2, -2], [-1, 4], 80, sqrt(80))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-3, 2], [-2, -2], [-1, 4], 80, sqrt(80)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
41. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[17, 0], [0, 1]] and x=[-2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, -3]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-2, -2], [-1, -3], [-1, 1], 18, sqrt(18))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-2, -2], [-1, -3], [-1, 1], 18, sqrt(18)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
42. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[18, 1], [1, 2]] and x=[-1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, -4]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-1, -1], [0, -4], [-1, 3], 30, sqrt(30))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-1, -1], [0, -4], [-1, 3], 30, sqrt(30)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
43. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, -5]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([0, 0], [1, -5], [-1, 5], 124, sqrt(124))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([0, 0], [1, -5], [-1, 5], 124, sqrt(124)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
44. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[20, 0], [0, 1]] and x=[1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, -6]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([1, 1], [2, -6], [-1, 7], 69, sqrt(69))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([1, 1], [2, -6], [-1, 7], 69, sqrt(69)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
45. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[21, 1], [1, 2]] and x=[2, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, -7]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([2, 2], [3, -7], [-1, 9], 165, sqrt(165))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([2, 2], [3, -7], [-1, 9], 165, sqrt(165)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
46. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[22, 2], [2, 5]] and x=[3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, -8]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([3, -2], [4, -8], [-1, 6], 178, sqrt(178))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([3, -2], [4, -8], [-1, 6], 178, sqrt(178)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
47. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[23, 0], [0, 1]] and x=[-3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, -9]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-3, -1], [5, -9], [-8, 8], 1536, sqrt(1536))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-3, -1], [5, -9], [-8, 8], 1536, sqrt(1536)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
48. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[24, 1], [1, 2]] and x=[-2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, -10]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-2, 0], [-5, -10], [3, 10], 476, sqrt(476))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-2, 0], [-5, -10], [3, 10], 476, sqrt(476)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
49. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[25, 2], [2, 5]] and x=[-1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, -11]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([-1, 1], [-4, -11], [3, 12], 1089, sqrt(1089))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([-1, 1], [-4, -11], [3, 12], 1089, sqrt(1089)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
50. Induced Distance Uses the Signed Vector Difference
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[26, 0], [0, 1]] and x=[0, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -12]. Give (x,y,difference=x−y,distance_square,exact_distance).
View answer and explanation
([0, 2], [-3, -12], [3, 14], 430, sqrt(430))
Compute the signed difference, its self-inner-product in the same matrix, then the nonnegative exact sqrt. Do not replace the induced metric with an unrelated coordinate norm. The exact result, retaining the requested order, is ([0, 2], [-3, -12], [3, 14], 430, sqrt(430)).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-distance-square · candidate
51. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[2, 0], [0, 1]] and x=[-3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 12]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(6, 22, 194, 36, 4268, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (6, 22, 194, 36, 4268, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
52. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[3, 1], [1, 2]] and x=[-2, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 11]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-16, 18, 202, 256, 3636, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-16, 18, 202, 256, 3636, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
53. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[4, 2], [2, 5]] and x=[-1, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, 10]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-8, 4, 416, 64, 1664, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-8, 4, 416, 64, 1664, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
54. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[5, 0], [0, 1]] and x=[0, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, 9]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(9, 1, 101, 81, 101, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (9, 1, 101, 81, 101, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
55. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[6, 1], [1, 2]] and x=[1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, 8]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(32, 18, 118, 1024, 2124, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (32, 18, 118, 1024, 2124, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
56. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[7, 2], [2, 5]] and x=[2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, 7]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-42, 32, 245, 1764, 7840, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-42, 32, 245, 1764, 7840, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
57. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, 6]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(18, 73, 44, 324, 3212, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (18, 73, 44, 324, 3212, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
58. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[9, 1], [1, 2]] and x=[-3, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, 5]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-69, 81, 106, 4761, 8586, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-69, 81, 106, 4761, 8586, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
59. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, 4]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-50, 37, 218, 2500, 8066, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-50, 37, 218, 2500, 8066, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
60. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, 3]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-38, 15, 185, 1444, 2775, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-38, 15, 185, 1444, 2775, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
61. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[12, 1], [1, 2]] and x=[0, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, 2]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-18, 8, 328, 324, 2624, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-18, 8, 328, 324, 2624, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
62. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[13, 2], [2, 5]] and x=[1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 1]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-58, 14, 310, 3364, 4340, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-58, 14, 310, 3364, 4340, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
63. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[14, 0], [0, 1]] and x=[2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 0]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-112, 56, 224, 12544, 12544, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-112, 56, 224, 12544, 12544, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
64. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[15, 1], [1, 2]] and x=[3, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -1]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-143, 143, 143, 20449, 20449, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-143, 143, 143, 20449, 20449, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
65. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[16, 2], [2, 5]] and x=[-3, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, -2]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(80, 140, 100, 6400, 14000, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (80, 140, 100, 6400, 14000, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
66. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[17, 0], [0, 1]] and x=[-2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, -3]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(40, 72, 26, 1600, 1872, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (40, 72, 26, 1600, 1872, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
67. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[18, 1], [1, 2]] and x=[-1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, -4]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(12, 22, 32, 144, 704, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (12, 22, 32, 144, 704, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
68. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, -5]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(0, 0, 124, 0, 0, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (0, 0, 124, 0, 0, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
69. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[20, 0], [0, 1]] and x=[1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, -6]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(34, 21, 116, 1156, 2436, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (34, 21, 116, 1156, 2436, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
70. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[21, 1], [1, 2]] and x=[2, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, -7]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(90, 100, 245, 8100, 24500, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (90, 100, 245, 8100, 24500, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
71. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[22, 2], [2, 5]] and x=[3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, -8]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(280, 194, 544, 78400, 105536, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (280, 194, 544, 78400, 105536, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
72. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[23, 0], [0, 1]] and x=[-3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, -9]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-336, 208, 656, 112896, 136448, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-336, 208, 656, 112896, 136448, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
73. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[24, 1], [1, 2]] and x=[-2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, -10]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(260, 96, 900, 67600, 86400, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (260, 96, 900, 67600, 86400, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
74. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[25, 2], [2, 5]] and x=[-1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, -11]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(59, 26, 1181, 3481, 30706, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (59, 26, 1181, 3481, 30706, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
75. A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[26, 0], [0, 1]] and x=[0, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -12]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).
View answer and explanation
(-24, 4, 378, 576, 1512, Yes, Cauchy_Schwarz_universal_certificate)
Calculate <x,y>, <x,x> and <y,y> exactly. Compare <x,y> squared with <x,x>*<y,y>. The signed first inner-product value is retained even though the inequality squares it. This instance checks consistency with the universal theorem; it does not establish the theorem by sampling. The exact result, retaining the requested order, is (-24, 4, 378, 576, 1512, Yes, Cauchy_Schwarz_universal_certificate).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-cauchy-schwarz · candidate
76. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[2, 0], [0, 1]] and x=[-3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 12]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-3, -2], [-5, 12], 432, 432, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-3, -2], [-5, 12], 432, 432, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
77. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[3, 1], [1, 2]] and x=[-2, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 11]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-2, -1], [-4, 11], 440, 440, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-2, -1], [-4, 11], 440, 440, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
78. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[4, 2], [2, 5]] and x=[-1, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, 10]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-1, 0], [-3, 10], 840, 840, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-1, 0], [-3, 10], 840, 840, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
79. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[5, 0], [0, 1]] and x=[0, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, 9]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([0, 1], [-2, 9], 204, 204, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([0, 1], [-2, 9], 204, 204, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
80. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[6, 1], [1, 2]] and x=[1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, 8]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([1, 2], [-1, 8], 272, 272, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([1, 2], [-1, 8], 272, 272, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
81. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[7, 2], [2, 5]] and x=[2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, 7]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([2, -2], [0, 7], 554, 554, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([2, -2], [0, 7], 554, 554, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
82. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, 6]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([3, -1], [1, 6], 234, 234, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([3, -1], [1, 6], 234, 234, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
83. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[9, 1], [1, 2]] and x=[-3, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, 5]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-3, 0], [2, 5], 374, 374, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-3, 0], [2, 5], 374, 374, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
84. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, 4]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-2, 1], [3, 4], 510, 510, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-2, 1], [3, 4], 510, 510, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
85. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, 3]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-1, 2], [4, 3], 400, 400, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-1, 2], [4, 3], 400, 400, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
86. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[12, 1], [1, 2]] and x=[0, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, 2]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([0, -2], [5, 2], 672, 672, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([0, -2], [5, 2], 672, 672, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
87. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[13, 2], [2, 5]] and x=[1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, 1]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([1, -1], [-5, 1], 648, 648, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([1, -1], [-5, 1], 648, 648, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
88. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[14, 0], [0, 1]] and x=[2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, 0]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([2, 0], [-4, 0], 560, 560, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([2, 0], [-4, 0], 560, 560, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
89. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[15, 1], [1, 2]] and x=[3, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -1]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([3, 1], [-3, -1], 572, 572, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([3, 1], [-3, -1], 572, 572, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
90. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[16, 2], [2, 5]] and x=[-3, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-2, -2]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-3, 2], [-2, -2], 480, 480, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-3, 2], [-2, -2], 480, 480, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
91. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[17, 0], [0, 1]] and x=[-2, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-1, -3]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-2, -2], [-1, -3], 196, 196, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-2, -2], [-1, -3], 196, 196, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
92. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[18, 1], [1, 2]] and x=[-1, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[0, -4]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-1, -1], [0, -4], 108, 108, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-1, -1], [0, -4], 108, 108, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
93. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, -5]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([0, 0], [1, -5], 248, 248, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([0, 0], [1, -5], 248, 248, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
94. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[20, 0], [0, 1]] and x=[1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[2, -6]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([1, 1], [2, -6], 274, 274, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([1, 1], [2, -6], 274, 274, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
95. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[21, 1], [1, 2]] and x=[2, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, -7]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([2, 2], [3, -7], 690, 690, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([2, 2], [3, -7], 690, 690, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
96. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[22, 2], [2, 5]] and x=[3, -2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[4, -8]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([3, -2], [4, -8], 1476, 1476, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([3, -2], [4, -8], 1476, 1476, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
97. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[23, 0], [0, 1]] and x=[-3, -1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[5, -9]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-3, -1], [5, -9], 1728, 1728, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-3, -1], [5, -9], 1728, 1728, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
98. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[24, 1], [1, 2]] and x=[-2, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-5, -10]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-2, 0], [-5, -10], 1992, 1992, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-2, 0], [-5, -10], 1992, 1992, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
99. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[25, 2], [2, 5]] and x=[-1, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-4, -11]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([-1, 1], [-4, -11], 2414, 2414, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([-1, 1], [-4, -11], 2414, 2414, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
100. Check the Parallelogram Identity in One Inner Product
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[26, 0], [0, 1]] and x=[0, 2]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[-3, -12]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
View answer and explanation
([0, 2], [-3, -12], 764, 764, Yes)
Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([0, 2], [-3, -12], 764, 764, Yes).
Reviewed step-by-step derivation
Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate
Canonical question JSONL · English JSONL · Validation report