{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0001","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fa606aebcacc3ed62c9b37ae33434b69bc04de5d0a134f21999b5c45f5d96d5","family_id":"qf-mod29-innernorm-square","seed":202610293001,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[2, 0], [0, 1]], [-3, -2], 22, sqrt(22), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0002","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"238eb6e5ac1f522638935be9ca0ef2c630f425187a708d39b45bd5760964b0b9","family_id":"qf-mod29-innernorm-square","seed":202610293002,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[3, 1], [1, 2]], [-2, -1], 18, sqrt(18), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0003","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01be3f61586159e03dc3ab3015652bfc48006893930e782819c5dd1bacdcce59","family_id":"qf-mod29-innernorm-square","seed":202610293003,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[4, 2], [2, 5]], [-1, 0], 4, sqrt(4), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0004","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eed855025b7fe38e34ce3cc7a04d8e9dddb0e77ac978c3758cd730c9a7ec6de","family_id":"qf-mod29-innernorm-square","seed":202610293004,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[5, 0], [0, 1]], [0, 1], 1, sqrt(1), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0005","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a38506fd7043ba57b1501b39d752912ed857e4638a9adb4324b9fd57fe02fc5","family_id":"qf-mod29-innernorm-square","seed":202610293005,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[6, 1], [1, 2]], [1, 2], 18, sqrt(18), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0006","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07c5c17587458aa067e5e336d14541fd65bac0c42119a18a933e436901b7c101","family_id":"qf-mod29-innernorm-square","seed":202610293006,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[7,2],[2,5]],\"x\":[2,-2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[7, 2], [2, 5]], [2, -2], 32, sqrt(32), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0007","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81d05f8f23e1386f92cc27a25102f018e5073caaea3b3bbad30e40a8447aeb35","family_id":"qf-mod29-innernorm-square","seed":202610293007,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0008","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f677b943101d60cc21802a52fa18b1f9589ad1c8992cc7960cff97dd7130412","family_id":"qf-mod29-innernorm-square","seed":202610293008,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[9, 1], [1, 2]], [-3, 0], 81, sqrt(81), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0009","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9983ac7366c0a05dd45502f77d5109f03ea4001f9307aa70cf21aad85b115af","family_id":"qf-mod29-innernorm-square","seed":202610293009,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[10, 2], [2, 5]], [-2, 1], 37, sqrt(37), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0010","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2911161b8f72f107ea8c0298e2783dbacca82e5c2ec8b789f81ae5b9da56eea9","family_id":"qf-mod29-innernorm-square","seed":202610293010,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [-1, 2], 15, sqrt(15), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0011","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b71400bce088b82a1209060bc959c1d690ef5b94c34fc7bf2739a5f4d858224","family_id":"qf-mod29-innernorm-square","seed":202610293011,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], [0, -2], 8, sqrt(8), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0012","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd62cadfe66c2245b709a428955f452aa469dd83a6ec4c0abf47e388b0b6e41f","family_id":"qf-mod29-innernorm-square","seed":202610293012,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [1, -1], 14, sqrt(14), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0013","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ac536796192f9ba60134f1675aa57ef9401ce5674666475bda6e6de16cd4b4","family_id":"qf-mod29-innernorm-square","seed":202610293013,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[14, 0], [0, 1]], [2, 0], 56, sqrt(56), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0014","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3db815d0b83414237c4c91a33e6997136538f7909582f70f49b4f2ed3eb6fb00","family_id":"qf-mod29-innernorm-square","seed":202610293014,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[15, 1], [1, 2]], [3, 1], 143, sqrt(143), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0015","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a7bcbe5e03ebe8c23934e02057837ccb9b965f20df9fca3553a5637eff3c1ff","family_id":"qf-mod29-innernorm-square","seed":202610293015,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[16, 2], [2, 5]], [-3, 2], 140, sqrt(140), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0016","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"951dcd07458f36ed3ff813439b366c496d140b371c0d6dc149eb6d9c53843659","family_id":"qf-mod29-innernorm-square","seed":202610293016,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[17, 0], [0, 1]], [-2, -2], 72, sqrt(72), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0017","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acf16028c30befdcb9b9c4e7a644df82df4f6c55e3d57bc79c5a4e661a6b2c81","family_id":"qf-mod29-innernorm-square","seed":202610293017,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[18, 1], [1, 2]], [-1, -1], 22, sqrt(22), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0018","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef109296ac6cc19c0fa491b67678700dd02497ca44d69fc9d1bfc2b9478aa8e1","family_id":"qf-mod29-innernorm-square","seed":202610293018,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[19, 2], [2, 5]], [0, 0], 0, sqrt(0), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0019","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3895964df8e931f5317439a464176237fb2f59b0ea4765dc0b4136fddffacd","family_id":"qf-mod29-innernorm-square","seed":202610293019,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[20, 0], [0, 1]], [1, 1], 21, sqrt(21), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0020","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e267cbb124d3c962a55ac80b7874ad71b4e54c1d73ff8509e6b2aa1bd36c009","family_id":"qf-mod29-innernorm-square","seed":202610293020,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[21,1],[1,2]],\"x\":[2,2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[21, 1], [1, 2]], [2, 2], 100, sqrt(100), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0021","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80bbdf398d9cc87290975fdbdab84aec7a5eb76706683514930e1971b75ad3aa","family_id":"qf-mod29-innernorm-square","seed":202610293021,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[22, 2], [2, 5]], [3, -2], 194, sqrt(194), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0022","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f420d464497344852b67a2261a396240f85a1f956650af01126b10c724d1cd6d","family_id":"qf-mod29-innernorm-square","seed":202610293022,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [-3, -1], 208, sqrt(208), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0023","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"073c0e8ac18998d193ed7a462edc26996aa4a33406b5f71ae755f88460eafa32","family_id":"qf-mod29-innernorm-square","seed":202610293023,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [-2, 0], 96, sqrt(96), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0024","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd279e1348d3c25e95075b2cbf14622f2cfee7b5c6365e716ae814deb0f2382f","family_id":"qf-mod29-innernorm-square","seed":202610293024,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[25,2],[2,5]],\"x\":[-1,1]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [-1, 1], 26, sqrt(26), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0025","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e992dab9bd44909a0659c876efb973c1d2d227869c8c44f4281c410b8c7718f","family_id":"qf-mod29-innernorm-square","seed":202610293025,"semantic_template":"qf-mod29-innernorm-square/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2]}"},"fields":{"title":"Use the Nonnegative Exact Root of the Same Self-Inner-Product","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([[26, 0], [0, 1]], [0, 2], 4, sqrt(4), same_inner_product_universal_norm_certificate)","explanation":"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)."},"skills":["Use the Nonnegative Exact Root of the Same Self-Inner-Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0026","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d5d64c3beede680c2745b424c4c5cce055bb84ab0d2933b51c8d9367c8436a","family_id":"qf-mod29-innernorm-distance-square","seed":202610293026,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-3, -2], [-5, 12], [2, -14], 204, sqrt(204))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0027","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"142d853ab14b724f64213764e4c456a895596ec54cb961a3cc234a0ff138ffd7","family_id":"qf-mod29-innernorm-distance-square","seed":202610293027,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-2, -1], [-4, 11], [2, -12], 252, sqrt(252))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0028","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf45d1b3244ac7d1fe7e1d02d8c9132288105a237fcc6696f4e7363190d26642","family_id":"qf-mod29-innernorm-distance-square","seed":202610293028,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-1, 0], [-3, 10], [2, -10], 436, sqrt(436))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0029","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18de182f6262bbe62a3714cffa42b9e44cf15477626af329503ea03c64216cde","family_id":"qf-mod29-innernorm-distance-square","seed":202610293029,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([0, 1], [-2, 9], [2, -8], 84, sqrt(84))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0030","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87dfaabc6bd8dd4704f8ed6d05a834580fe12ee93b40a6e6a694c759d0d2aa9d","family_id":"qf-mod29-innernorm-distance-square","seed":202610293030,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([1, 2], [-1, 8], [2, -6], 72, sqrt(72))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0031","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f405c5a8d47d73d4f7d79a392e001adbdfcf394e0a94f332ba5853dd92849bf1","family_id":"qf-mod29-innernorm-distance-square","seed":202610293031,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([2, -2], [0, 7], [2, -9], 361, sqrt(361))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0032","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b33e9fdc32efba455288d1224615f59de10501fe893eaba55c2db95af34f75ad","family_id":"qf-mod29-innernorm-distance-square","seed":202610293032,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([3, -1], [1, 6], [2, -7], 81, sqrt(81))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0033","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ca7c1b78489f8c760a53df48aea78b461a7c6b9ff104e610fc642d9d0cac4c","family_id":"qf-mod29-innernorm-distance-square","seed":202610293033,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-3, 0], [2, 5], [-5, -5], 325, sqrt(325))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0034","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43c32909ea0555b02a6f5fe99e05edebd68c1a0bac8694c38a65a57c2269fc6","family_id":"qf-mod29-innernorm-distance-square","seed":202610293034,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-2, 1], [3, 4], [-5, -3], 355, sqrt(355))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0035","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47799801a0edc9727c3084521199db2642b074b6b0f313caf7dd70ddae432f50","family_id":"qf-mod29-innernorm-distance-square","seed":202610293035,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-1, 2], [4, 3], [-5, -1], 276, sqrt(276))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0036","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1ab47988521b3ddb92a304b689a9e7e6e1e8eb130a4ed8144f809b002904284","family_id":"qf-mod29-innernorm-distance-square","seed":202610293036,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([0, -2], [5, 2], [-5, -4], 372, sqrt(372))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0037","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cf6938b6ee8b0e5c58063cbaee4d066bee55a2bff048eb63a7dca4dfee78e7b","family_id":"qf-mod29-innernorm-distance-square","seed":202610293037,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([1, -1], [-5, 1], [6, -2], 440, sqrt(440))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0038","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06e922eada4d0ecef504bfe8deb0ae7a7839971061a2cf596ffc54ef93469e53","family_id":"qf-mod29-innernorm-distance-square","seed":202610293038,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([2, 0], [-4, 0], [6, 0], 504, sqrt(504))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0039","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a08c4fe44f2d9c4f3616d071b8200899a4f2e34031b2c069c47e5267263b9a88","family_id":"qf-mod29-innernorm-distance-square","seed":202610293039,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([3, 1], [-3, -1], [6, 2], 572, sqrt(572))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0040","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dc795ac6d104e18829cfcc8fdee9a670dfb3404cb8fb53abaadd4ad833bac19","family_id":"qf-mod29-innernorm-distance-square","seed":202610293040,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-3, 2], [-2, -2], [-1, 4], 80, sqrt(80))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0041","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d43d0e4a130bd11111168b1184ebf994b89568962e8122cf3503fd86c91202ad","family_id":"qf-mod29-innernorm-distance-square","seed":202610293041,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-2, -2], [-1, -3], [-1, 1], 18, sqrt(18))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0042","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca3b487b2330c950096fffd2b5f42436179f402bbcf3e49b6dcde6cd8e3f0bc8","family_id":"qf-mod29-innernorm-distance-square","seed":202610293042,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-1, -1], [0, -4], [-1, 3], 30, sqrt(30))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0043","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e416186c6d1262256979e6308cf6de96147ece33e6d4d2e4db772496d2bd5c6b","family_id":"qf-mod29-innernorm-distance-square","seed":202610293043,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([0, 0], [1, -5], [-1, 5], 124, sqrt(124))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0044","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ff63b2651c7224c33ebe8e99aa82d5d589cca4bca8fd14c8e79391cd6a13b06","family_id":"qf-mod29-innernorm-distance-square","seed":202610293044,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([1, 1], [2, -6], [-1, 7], 69, sqrt(69))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0045","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815f2a2c12f629a7aa46e64222ee993d1263e511a16f53a99bdaadf14e55c72f","family_id":"qf-mod29-innernorm-distance-square","seed":202610293045,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([2, 2], [3, -7], [-1, 9], 165, sqrt(165))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0046","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80c2d1622e6ad74c016f5db8e238dee1cca38c1cd600e987ef9716674b0c9ce7","family_id":"qf-mod29-innernorm-distance-square","seed":202610293046,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([3, -2], [4, -8], [-1, 6], 178, sqrt(178))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0047","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39f5830e7ebb5084a4c9df0e3c28859ad972df1947834df30b762da89700c2b5","family_id":"qf-mod29-innernorm-distance-square","seed":202610293047,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-3, -1], [5, -9], [-8, 8], 1536, sqrt(1536))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0048","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa911cd427e0b576b1a11b6cf01cdc7e2a3a7a98addeda177b2c078512f929ec","family_id":"qf-mod29-innernorm-distance-square","seed":202610293048,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-2, 0], [-5, -10], [3, 10], 476, sqrt(476))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0049","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c80c682fdd6f6743ddf8946bf80bb77315a8d0a89fb255821a8405d3033c4a5","family_id":"qf-mod29-innernorm-distance-square","seed":202610293049,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([-1, 1], [-4, -11], [3, 12], 1089, sqrt(1089))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0050","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbf1ec360d444d6547475e5cd2dbf95067515bd73949b80f03c9006a346242a9","family_id":"qf-mod29-innernorm-distance-square","seed":202610293050,"semantic_template":"qf-mod29-innernorm-distance-square/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}"},"fields":{"title":"Induced Distance Uses the Signed Vector Difference","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"([0, 2], [-3, -12], [3, 14], 430, sqrt(430))","explanation":"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))."},"skills":["Induced Distance Uses the Signed Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0051","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8c612d945cce318cb939926ec04145a9c1068fd96883a9bbdc49bfb71c92172","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293051,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(6, 22, 194, 36, 4268, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0052","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f5b0a9a41a83bf6f57b3633896dd2cc2f83fab4a5f9beedb9e9c850f461b94f","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293052,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-16, 18, 202, 256, 3636, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0053","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8c96173dc53e78e595a45e0afda870e35c5848a69e28a735d3de42cd7570209","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293053,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-8, 4, 416, 64, 1664, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0054","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b53ac366e397f0f80c717616aafbf19febf61b8ce7441cd1b235bf4e1427f9","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293054,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(9, 1, 101, 81, 101, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0055","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8379e8a3abbf15082d4b585613d68fa8d286ef4a1f755caac5adaa18daea2c93","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293055,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(32, 18, 118, 1024, 2124, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0056","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25cd749b762cbeb5683f7325f28a07c4ba001206adf05418d1503f73edd13dff","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293056,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-42, 32, 245, 1764, 7840, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0057","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"648a9f1ea5b353b8857bd9728f395f21dbdc5931482eb0e523addf185c7d45b7","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293057,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(18, 73, 44, 324, 3212, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0058","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4ee20bbdb4297e3361ef4c963443fb198ff476672cdf45efadff397611f1eba","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293058,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-69, 81, 106, 4761, 8586, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0059","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04a9b968ee4dc76629d168d3d036c6d1d81674a048acf1361a8a90d6fbdb82f9","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293059,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-50, 37, 218, 2500, 8066, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0060","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cc0f3b11a4878d9e32e64e24c35bd089ea1543c21429d248d800bb319f6ffeb","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293060,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-38, 15, 185, 1444, 2775, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0061","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf835d6f2ff9108be8ef4b068399545ed1c0a00fa9076433f08ccd4ed980e6a","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293061,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-18, 8, 328, 324, 2624, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0062","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0a6c7340e8300d7379eba39fa94090e266e43b6c01d0b696212526cd0c8d3e4","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293062,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-58, 14, 310, 3364, 4340, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0063","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e14576cae16827e2631ca07f97350367a7159dcc6601be335551dc25abb08e2b","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293063,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-112, 56, 224, 12544, 12544, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0064","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14941f9acfcdf651413f08c20e9c91681b28822b60a4b6ab4397546871892401","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293064,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-143, 143, 143, 20449, 20449, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0065","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb2df6574398dc844c581b04d62cdb7dcbf4fa8c8c04883d349fc64c2432d822","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293065,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(80, 140, 100, 6400, 14000, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0066","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2630a54b5096b1874e7022e5b3bd5292583ecc86397c6937f7b6fc747b94a590","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293066,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(40, 72, 26, 1600, 1872, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0067","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4151004cef2fbf50a91eb7ee95afd8aca870510ac1e4ab9f0a4edd2a316808f8","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293067,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(12, 22, 32, 144, 704, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0068","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"959a8ec1070e280b4a24cfa7a1aa22f305b0ee563ec9e8c67fcacf7b72251f1b","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293068,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(0, 0, 124, 0, 0, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0069","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adb0fec9acdb687258f69e5d82dc04c09105f57c2dd9936e5a0ae1e4c7533ff1","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293069,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(34, 21, 116, 1156, 2436, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0070","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee15b5bcbef3406a808bdddb3cccb89370fefa20b20af42f33160aa5818aa97","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293070,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(90, 100, 245, 8100, 24500, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0071","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b281e3fd6c38a8ea7a95190050cc427dbed6121f2d1081ca30ea15b74d67be8","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293071,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(280, 194, 544, 78400, 105536, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0072","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d9897266b467b5d01bb92df8f0eae991a333a73a188d34d7d10f828a7f4d082","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293072,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-336, 208, 656, 112896, 136448, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0073","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe018e2f85fa45453c1a51c9a4b7de0a6a8c84d8f4a6418033c1cc6f34dfe1c","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293073,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(260, 96, 900, 67600, 86400, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0074","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8ec13094b55d8cf412a21f985ab159ede5f5cf2f5b317dcc20230c1ba344ffe","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293074,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(59, 26, 1181, 3481, 30706, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0075","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a22d676e010d03c18d9f62f367b291f2bd1e7bed830f1e8957df506990745c3e","family_id":"qf-mod29-innernorm-cauchy-schwarz","seed":202610293075,"semantic_template":"qf-mod29-innernorm-cauchy-schwarz/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}"},"fields":{"title":"A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary","stem":"Use the same certified real inner product on R², =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(), 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).","answer":{"kind":"expression","value":"(-24, 4, 378, 576, 1512, Yes, Cauchy_Schwarz_universal_certificate)","explanation":"Calculate , and exactly. Compare squared with *. 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)."},"skills":["A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0076","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7712c09e7d7ebf6c55f22f9b059dc51e59f523c67413de619560fa3d3b738c71","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293076,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[2,0],[0,1]],\"x\":[-3,-2],\"y\":[-5,12]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-3, -2], [-5, 12], 432, 432, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0077","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4debde77a75ce6018e2764ede129747f527607c5e7486140e81f70aacf03f5e9","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293077,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[3,1],[1,2]],\"x\":[-2,-1],\"y\":[-4,11]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-2, -1], [-4, 11], 440, 440, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0078","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a643d43fe06f4ce51656cdaf4331f859531de4a5ba2818c4b0fc7a3977d3bbe7","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293078,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[4,2],[2,5]],\"x\":[-1,0],\"y\":[-3,10]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-1, 0], [-3, 10], 840, 840, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0079","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"081d8bf395b273280b7f1aa3c8874fd88549a6273dbd8955f7672ae34cc27b2b","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293079,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[5,0],[0,1]],\"x\":[0,1],\"y\":[-2,9]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([0, 1], [-2, 9], 204, 204, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0080","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ca96c74ac18ca113436b59a53f0a640561b4671eaf11b3fba7c6a37b18e76a","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293080,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[-1,8]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([1, 2], [-1, 8], 272, 272, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0081","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f728c656afd75e656a4dba5fc4ffeb45b9ccb2876fac1cf7d2982e277b26a2d","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293081,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[7,2],[2,5]],\"x\":[2,-2],\"y\":[0,7]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([2, -2], [0, 7], 554, 554, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0082","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60c7a9e1d45244883443e58f01ba80be8b865435bafeec7832bd663de14cfd33","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293082,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[8,0],[0,1]],\"x\":[3,-1],\"y\":[1,6]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([3, -1], [1, 6], 234, 234, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0083","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93671210cddca1922772f3051396f73aece38d79d61c9f8cea4caa4bf7dac57c","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293083,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[9,1],[1,2]],\"x\":[-3,0],\"y\":[2,5]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-3, 0], [2, 5], 374, 374, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0084","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0335e82667e39b3cafd44347ea18ec6ed10c7f65eb575f70d2d59a5f40d1e2db","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293084,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[10,2],[2,5]],\"x\":[-2,1],\"y\":[3,4]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-2, 1], [3, 4], 510, 510, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0085","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"470e57c5cb23a2e2323a9c37369df36d2558854edcca3f103d2ef703cceb9dbc","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293085,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[11,0],[0,1]],\"x\":[-1,2],\"y\":[4,3]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-1, 2], [4, 3], 400, 400, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0086","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d53946813a18d5e22723cfe9304c847856f6d997f0a854e98e82f40618e1cbd5","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293086,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[12,1],[1,2]],\"x\":[0,-2],\"y\":[5,2]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([0, -2], [5, 2], 672, 672, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0087","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a874bef202bcbdf47b7e32d24fa202da30c3aa76edc55e90435cd583553a6162","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293087,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[-5,1]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([1, -1], [-5, 1], 648, 648, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0088","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62d1aea8ddcec43f23160376a0a6e56099d1bf8171f1cc5171cb94dc98bd5046","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293088,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[14,0],[0,1]],\"x\":[2,0],\"y\":[-4,0]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([2, 0], [-4, 0], 560, 560, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0089","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce9a15a52b4900012ed47460a4be5b6eaee56f9d2a8a23c5cefb76e362083c3","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293089,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[15,1],[1,2]],\"x\":[3,1],\"y\":[-3,-1]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([3, 1], [-3, -1], 572, 572, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0090","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d4e44c398cb33db7112e8372b38faaadc77041c77df9c3f91b700ec2ea303da","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293090,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[16,2],[2,5]],\"x\":[-3,2],\"y\":[-2,-2]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-3, 2], [-2, -2], 480, 480, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0091","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a2bde515b576f7f4b0605bb3c65e799d9fb73563643a233b52a1986fdecd03","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293091,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[17,0],[0,1]],\"x\":[-2,-2],\"y\":[-1,-3]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-2, -2], [-1, -3], 196, 196, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0092","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bcc335f1d50507d8ab5c6469883554ad6df5018c2842dd4a299cebef37c509e","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293092,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[18,1],[1,2]],\"x\":[-1,-1],\"y\":[0,-4]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-1, -1], [0, -4], 108, 108, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0093","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c13fc1e646bf7f13de812d5bfbee5d1528124c92f884854cce8be65ea7a9d84","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293093,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([0, 0], [1, -5], 248, 248, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0094","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56a6826867ac936d869b1101a4ad028fabce02b806a35a1ab161798475b4ce25","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293094,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([1, 1], [2, -6], 274, 274, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0095","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e7cd8afd44a6be7caef97ab3ef88e183e3503054da7639e07e7e22a78ba48a4","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293095,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[21,1],[1,2]],\"x\":[2,2],\"y\":[3,-7]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([2, 2], [3, -7], 690, 690, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0096","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c586a54419a36c5bfa68c4ecd328bf1117d54a3f08ef5f52da0abd9bbf4b01e1","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293096,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[22,2],[2,5]],\"x\":[3,-2],\"y\":[4,-8]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([3, -2], [4, -8], 1476, 1476, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0097","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c99244fda374226d4f7d3626781cc6df4b4fb51df2294a6cc66b66aba1289ba","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293097,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[23,0],[0,1]],\"x\":[-3,-1],\"y\":[5,-9]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-3, -1], [5, -9], 1728, 1728, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0098","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f91f65e271cdda41fff221b14dd5af802fe00918cbc26166c2454a2ed7db5716","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293098,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[24,1],[1,2]],\"x\":[-2,0],\"y\":[-5,-10]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-2, 0], [-5, -10], 1992, 1992, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0099","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc16b73fabb7afb3d2e283117434a309420f32f891d1376dfdccf4a685b173ac","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293099,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[25,2],[2,5]],\"x\":[-1,1],\"y\":[-4,-11]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([-1, 1], [-4, -11], 2414, 2414, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r03-inner-product-induced-norm-0100","batch_id":"batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f88455d3ac417b6934e79968d1f13f32c6d7b371d3e3de3b9b199fea4731878","family_id":"qf-mod29-innernorm-parallelogram","seed":202610293100,"semantic_template":"qf-mod29-innernorm-parallelogram/{\"matrix\":[[26,0],[0,1]],\"x\":[0,2],\"y\":[-3,-12]}"},"fields":{"title":"Check the Parallelogram Identity in One Inner Product","stem":"Use the same certified real inner product on R², =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(), 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?).","answer":{"kind":"expression","value":"([0, 2], [-3, -12], 764, 764, Yes)","explanation":"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)."},"skills":["Check the Parallelogram Identity in One Inner Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}