# Exact Induced Norms, Distances, Cauchy–Schwarz and Parallelogram Checks

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r03-inner-product-induced-norm-0100/)

Inner-product-induced norm · qf-mod29-innernorm-parallelogram · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations/validation-report.json)