{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r02-real-hilbert-space-0001","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a1598de636d8f34a75d22841902f2768b1f470940e4f840db72af93474d6975","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302001,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[2,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0002","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"260edc37a2b6a825ca0be8165cd0ae03bc0ccd49135ea07922c5d771e7c3d942","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302002,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[3,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0003","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ec501ae6b9ec054e19a8461cbc8e3bc2bc213fba55c178e3a03156a655b8de","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302003,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[4,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0004","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b3bf2e3ad39ccd2a7b3e774cb2fba9b62e73ef6c33cc15c30737ed47a06a29c","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302004,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[5,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0005","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e9ea0a708647fb170a58b8608c69734c82fcfc36540034c5a838f36e41c17f2","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302005,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[6,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0006","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6766b8db8b0306d2394b2abd026b319ff37f29b6c8b8dc8ec776a0ed4cb474b","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302006,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[7,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0007","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8759ff7bdaf32f7174546fc9adb743f65ed9b73dfd949ef5f33e8878b43b6ba","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302007,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[8,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0008","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6d0c13be09b1e4c35058d4cf02057f1fb58cd4faaac1e6db6a4a08fbe907d75","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302008,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[9,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0009","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1efca9ec182117020b253e10cab8390db5f591653aad24d007720ffdf9329e30","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302009,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[10,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0010","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6570e370074cca0ae34d9d0a479a36f333245f9f96446128cfadb08ecb4663","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302010,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[11,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0011","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5915edf811aef1d66d523aacac74a6ef10aae726c36a657989c02a81c9b4f1","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302011,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[12,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0012","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56fd9ba4a406bf012272f85d161a90253dc8964917fa0864bd462bd5ddf1b8ba","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302012,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[13,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0013","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3443170d462860e7b6a5492bd5c8623bb07e76fae109dfc242b26d9d9874a1bd","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302013,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[14,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0014","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94a7e9e63110c71c817844476f3d5f256409a27dd09946b58d2b9847ec33956d","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302014,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[15,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0015","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7b1e2ff6b9ae41a9cf7d3cdbfdbc2ef0aa5dbfb7ac88d48bea112eaa8cf534","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302015,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[16,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0016","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb0bb8f0aee9aa845630cceee5be9b79d91383c1c347212fc822a5a8edb693d0","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302016,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[17,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0017","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"249d8d6c207825af2f27befdc1c01e544d6f8b3d535fc7d6795c432a4afa1892","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302017,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[18,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0018","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a65f7dca4ddea74fc2fc4c5c33ae827e2ed27d018d4d2db0a041f1b4371f94a","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302018,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[19,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0019","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89918f949f58e709c590b7a1900a23b0ae44ab568e362b28fb096c47bfd1f4e","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302019,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[20,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0020","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808cf4104394685f4654f18ab2dddf09dd350c163a1f5e451361504745e59776","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302020,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[21,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0021","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f97a2b800fac66c9e99f00bb1dc1fe2795808d7bb51f8852fb3abab4fc85adf","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302021,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[22,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0022","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4fca2e0e3c2f194aaf9d145e8150f6408684acdd8b40caaa0a7e4431012a92a","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302022,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[23,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0023","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d1a5bb21c062699591fe94ee515f86a340fb5458241a908e8222c8d183d6b9e","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302023,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[24,1],[1,2]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0024","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2c81c6f8fdf113aeabc48ee4df0fed8f9721f7cbb38fbafa17ec802e64edc8f","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302024,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[25,2],[2,5]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0025","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"715d180b93a1737725664b2051dacfee863ea5e6dea1b1716bee5332bf9d6a65","family_id":"qf-mod30-hilbert-named-certificate","seed":202610302025,"semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[26,0],[0,1]]}"},"fields":{"title":"Bind Finite-Dimensional Completeness to the Same Inner-Product Space","stem":"Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).","answer":{"kind":"expression","value":"(2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)","explanation":"The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim)."},"skills":["Bind Finite-Dimensional Completeness to the Same Inner-Product Space"],"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-20261030-r02-real-hilbert-space-0026","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0785bf6e69c75d50c330703be51f64d576b283c9e6bf33bed7fc5c1840ee93fb","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302026,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[2,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0027","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d943508eb587a3d3da13a981782b977e510b6e62c1951a2486363fa1ed5844b","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302027,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[3,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0028","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc57490edc22215f06239e18da434385732df0813f6ce52a2c830df5deaf4f40","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302028,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[4,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0029","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aca17ebab5d163ee1d31f29e07a8e9f4cda1267184fee7f35ab04e72a17657f0","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302029,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[5,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0030","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8ac545e77a5f468e27301c0b07688d50a286e65613f5683629ee9425ef6e4ca","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302030,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[6,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0031","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38bd180885cc966e79463e1abba7cc13e2a88b8764d5a44bc6261668a6774ae1","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302031,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[7,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0032","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ea44446d89597741eea54583e9dbbdd364fde3454ee07c2550c99c1ebbee48d","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302032,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[8,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0033","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e7230b83e633aba709e191226c93bf38b02b2598c595fdd8a039cf2144c98ee","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302033,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[9,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0034","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4fd8df53955e339aa577fed27eacf44b8fd5b6e309a35241c1c4153ff7ee317","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302034,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0035","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6bc5de0517b9456eb2aa420a646820761bd3d749b24eb102cde22ab88d35fa4","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302035,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[11,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0036","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e61a8e137bd996f447194e63c7498e659b854dcb56c9c02bd693914b6e0b6d","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302036,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[12,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, No)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, No)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0037","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"517ea70de6d97b08d83f9049d256c10205153caa117e61307071e4dd0382bbb3","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302037,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[13,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0038","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d4162ad072dff518189dd2d64f3c23639c651d9e92b293b03f9e731cbe4ff4","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302038,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[14,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0039","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f79472b8f9e903218b019a899b1cc073a66d016539d1855eccd6be715fea5fc9","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302039,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[15,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0040","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25cf897ab1e92eaa568bfeb4d44b5310be2e4c453dee31bea4670e9c8e231bee","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302040,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[16,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0041","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a409ede137b99f76413d03287ba8fd3d5e22054b8feb62042083043930395feb","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302041,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[17,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0042","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a012d0dffef5f194bfe244929866c555c138f90676d596db96d6db78a6a01c62","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302042,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[18,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0043","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83f2c6d8debf44df9d8ce0917dbbe5267e6235901d9a65de8a14268b30a59a2","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302043,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[19,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0044","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac8edee0985dc29e37ad839a82c2498bff3a0e1ddcf980062f97bc8c08be751","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302044,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[20,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0045","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36937f750771cd776d894096dd5095854ecd4ddb6cbcdb9cb8695c6559439daa","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302045,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[21,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0046","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2279aea77fd0a4898b2423dd53a2ff20e64fe3f179871d5bb4955ce57a3c8902","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302046,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[22,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0047","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11834cd75b835bdc3c559c21a247f16bb15e538ec7bf63964c320c1c4aac479c","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302047,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[23,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0048","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c8ef076f72dca3b01e70ec64788ba7d37fbdc2a6e94ea028c927f3860409bb5","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302048,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[24,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0049","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d3105742b5fa5f5c6d0eb004491862056c516ec24eeba37e253c41c51632065","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302049,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[25,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0050","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b3a37b193e7870517e59eeaf6914af578e2330b0d9c648e03c1b48d4425a150","family_id":"qf-mod30-hilbert-component-checklist","seed":202610302050,"semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[26,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"},"fields":{"title":"A Completeness Checklist Must Use One Carrier, Norm and Metric","stem":"Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).","answer":{"kind":"expression","value":"(2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)","explanation":"The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes)."},"skills":["A Completeness Checklist Must Use One Carrier, Norm and Metric"],"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-20261030-r02-real-hilbert-space-0051","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13a3e9b1b52d312c05a986bfe845ab66881892998f5a991bc70beafcb3031c93","family_id":"qf-mod30-hilbert-distance","seed":202610302051,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-12,-2],\"matrix\":[[2,0],[0,1]],\"right\":[12,-3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-12, -2], right vector=[12, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0052","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ab0e7728adbc098644a261798be44682d27d2f5203b92e0099ef6af5c962539","family_id":"qf-mod30-hilbert-distance","seed":202610302052,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-11,-1],\"matrix\":[[3,1],[1,2]],\"right\":[11,-2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-11, -1], right vector=[11, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0053","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe478c24cf53df1bacaf29103001af9a1aa4229b5727ff7f2fbaab4bd35b56e3","family_id":"qf-mod30-hilbert-distance","seed":202610302053,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-10,0],\"matrix\":[[4,2],[2,5]],\"right\":[10,-1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-10, 0], right vector=[10, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0054","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e5cfa70934c82637236fdb014bb475c5f98215dfa1992e4d352afddf9dfcbfa","family_id":"qf-mod30-hilbert-distance","seed":202610302054,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-9,1],\"matrix\":[[5,0],[0,1]],\"right\":[9,0]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-9, 1], right vector=[9, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0055","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"024e4d2d67a1cb1c2f79c77b50b4ae445deb434903797e3e3640fef4f7f45f16","family_id":"qf-mod30-hilbert-distance","seed":202610302055,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-8,2],\"matrix\":[[6,1],[1,2]],\"right\":[8,1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-8, 2], right vector=[8, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0056","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34fa0f6b8f836bdc643051f691894c6afa9f9d1dd0da405f1b75f1c4a01d2474","family_id":"qf-mod30-hilbert-distance","seed":202610302056,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-7,-2],\"matrix\":[[7,2],[2,5]],\"right\":[7,2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-7, -2], right vector=[7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0057","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8776081d7c2e3370ec748b065b1c34e8620a85c5c0e63124146a8cb6b549fc9f","family_id":"qf-mod30-hilbert-distance","seed":202610302057,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-6,-1],\"matrix\":[[8,0],[0,1]],\"right\":[6,3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-6, -1], right vector=[6, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0058","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d68faeb1502ccf7c960f885066a683ed470a2e76a62b0181edf9a7fe362a111c","family_id":"qf-mod30-hilbert-distance","seed":202610302058,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-5,0],\"matrix\":[[9,1],[1,2]],\"right\":[5,-3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-5, 0], right vector=[5, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-5, 0], [5, -3], [-10, 3], 858, sqrt(858))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-5, 0], [5, -3], [-10, 3], 858, sqrt(858))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0059","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9cf7df1f34828cfb1c8994d22e5cd4bc3b6afcaa9520e49cf455eeccdb59538","family_id":"qf-mod30-hilbert-distance","seed":202610302059,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-4,1],\"matrix\":[[10,2],[2,5]],\"right\":[4,-2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-4, 1], right vector=[4, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-4, 1], [4, -2], [-8, 3], 589, sqrt(589))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-4, 1], [4, -2], [-8, 3], 589, sqrt(589))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0060","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8475157de0c4e1174c8af0facdd4f580026f81bcac001acb0c861df0678f0b4","family_id":"qf-mod30-hilbert-distance","seed":202610302060,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-3,2],\"matrix\":[[11,0],[0,1]],\"right\":[3,-1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-3, 2], right vector=[3, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-3, 2], [3, -1], [-6, 3], 405, sqrt(405))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-3, 2], [3, -1], [-6, 3], 405, sqrt(405))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0061","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b05eca952eb1fee8cda3b89d6ac67ee813151ab28240b126297e52d42aa703","family_id":"qf-mod30-hilbert-distance","seed":202610302061,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-2,-2],\"matrix\":[[12,1],[1,2]],\"right\":[2,0]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-2, -2], right vector=[2, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-2, -2], [2, 0], [-4, -2], 216, sqrt(216))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-2, -2], [2, 0], [-4, -2], 216, sqrt(216))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0062","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd41f8549e7c7760f8a8e7162c6cc248317ca25c02447360dffcb11358565844","family_id":"qf-mod30-hilbert-distance","seed":202610302062,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-1,-1],\"matrix\":[[13,2],[2,5]],\"right\":[1,1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-1, -1], right vector=[1, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([-1, -1], [1, 1], [-2, -2], 88, sqrt(88))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-1, -1], [1, 1], [-2, -2], 88, sqrt(88))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0063","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c052c87e23f94159ca8ac66a1e5d37f55495ff04c58868a8203fd330f57d4e8","family_id":"qf-mod30-hilbert-distance","seed":202610302063,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[0,0],\"matrix\":[[14,0],[0,1]],\"right\":[0,2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[0, 0], right vector=[0, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([0, 0], [0, 2], [0, -2], 4, sqrt(4))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([0, 0], [0, 2], [0, -2], 4, sqrt(4))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0064","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33490325c525218adbbe59e04b26bbefd42f52111eff92992ac4f50fbf992dc5","family_id":"qf-mod30-hilbert-distance","seed":202610302064,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[1,1],\"matrix\":[[15,1],[1,2]],\"right\":[-1,3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[1, 1], right vector=[-1, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([1, 1], [-1, 3], [2, -2], 60, sqrt(60))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([1, 1], [-1, 3], [2, -2], 60, sqrt(60))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0065","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bacd7467af1b081e485f2c038d45039255d38aae114d78c4be22318ba886ae66","family_id":"qf-mod30-hilbert-distance","seed":202610302065,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[2,2],\"matrix\":[[16,2],[2,5]],\"right\":[-2,-3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[2, 2], right vector=[-2, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([2, 2], [-2, -3], [4, 5], 461, sqrt(461))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([2, 2], [-2, -3], [4, 5], 461, sqrt(461))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0066","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e58b80e7138840aaa2a62b26a284062f263f2260d94f09393c422a4deccf9c8","family_id":"qf-mod30-hilbert-distance","seed":202610302066,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[3,-2],\"matrix\":[[17,0],[0,1]],\"right\":[-3,-2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[3, -2], right vector=[-3, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([3, -2], [-3, -2], [6, 0], 612, sqrt(612))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([3, -2], [-3, -2], [6, 0], 612, sqrt(612))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0067","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e860a7ccc6819dabdca93d4c9d4d94d443a64c6b1ee0a0120f6aceee5a4b2b17","family_id":"qf-mod30-hilbert-distance","seed":202610302067,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[4,-1],\"matrix\":[[18,1],[1,2]],\"right\":[-4,-1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[4, -1], right vector=[-4, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0068","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bbfcbd0dab8f2dac8ee6b100372db87271c36ef70dcb829989a310537cc6215","family_id":"qf-mod30-hilbert-distance","seed":202610302068,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[5,0],\"matrix\":[[19,2],[2,5]],\"right\":[-5,0]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[5, 0], right vector=[-5, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0069","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ef0fff5196164a386e87dbca3d640c1d86f1c526d765faf081887214d87a6bd","family_id":"qf-mod30-hilbert-distance","seed":202610302069,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[6,1],\"matrix\":[[20,0],[0,1]],\"right\":[-6,1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[6, 1], right vector=[-6, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0070","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb88cf5abe91e75527c87f8710b0ea919d788b1ba77f436976fcff996561ad72","family_id":"qf-mod30-hilbert-distance","seed":202610302070,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[7,2],\"matrix\":[[21,1],[1,2]],\"right\":[-7,2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[7, 2], right vector=[-7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0071","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e0fa236e96675b85a78fcb132cd3d1a6c6b74f1de2c2e1ca8a3f921a4b1d6b8","family_id":"qf-mod30-hilbert-distance","seed":202610302071,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[8,-2],\"matrix\":[[22,2],[2,5]],\"right\":[-8,3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[8, -2], right vector=[-8, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0072","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46b660136634c3884177adfa19545e176fb69dd926c40ee7ae48eac58afd1d72","family_id":"qf-mod30-hilbert-distance","seed":202610302072,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[9,-1],\"matrix\":[[23,0],[0,1]],\"right\":[-9,-3]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[9, -1], right vector=[-9, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0073","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0dc22e0011ab7b530d2193e3ada6aa53887e7845034cdcc809b9a9e7823a8a8","family_id":"qf-mod30-hilbert-distance","seed":202610302073,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[10,0],\"matrix\":[[24,1],[1,2]],\"right\":[-10,-2]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[10, 0], right vector=[-10, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0074","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c339ea9a0ac2a04ac0423f9af13e19ddeded7c2b302d4aa78a45f5b45b1130","family_id":"qf-mod30-hilbert-distance","seed":202610302074,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[11,1],\"matrix\":[[25,2],[2,5]],\"right\":[-11,-1]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[11, 1], right vector=[-11, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0075","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7fccddf4e41dd9065977282a5a5e00c0cfcef8ec69188afceb878da0d1cb5f","family_id":"qf-mod30-hilbert-distance","seed":202610302075,"semantic_template":"qf-mod30-hilbert-distance/{\"left\":[12,2],\"matrix\":[[26,0],[0,1]],\"right\":[-12,0]}"},"fields":{"title":"Compute Distance in the Norm Induced by This Positive-Definite Form","stem":"Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[12, 2], right vector=[-12, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).","answer":{"kind":"expression","value":"([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980))","explanation":"Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980))."},"skills":["Compute Distance in the Norm Induced by This Positive-Definite Form"],"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-20261030-r02-real-hilbert-space-0076","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43f15b97f0e8ea8506359db50b4cf26cb27de80f3fb500f03a37ffd55b815153","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302076,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[2,0],[0,1]],\"x\":[1,-2],\"y\":[0,-2]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[0, -2]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0077","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8c8d651bede5f5fb68d872c21a1c51a1b21db7361d70fc58ecbb2174df5c74","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302077,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[3,1],[1,2]],\"x\":[1,-1],\"y\":[1,-3]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[1, -3]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0078","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cbe58c84d1b8efa8ec30b95fed938cc59be0a7b8e85f65ee189c85df280806f","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302078,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[4,2],[2,5]],\"x\":[1,0],\"y\":[2,-4]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[2, -4]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 0], [2, -4], 0, 68, 68, Yes)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [2, -4], 0, 68, 68, Yes)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0079","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"312250a924befb6d6a2fbc0855cb6725cf48f9471e804e779dcc565ce481fb58","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302079,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[5,0],[0,1]],\"x\":[1,1],\"y\":[0,-5]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[0, -5]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0080","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95538ef586d4aad08515b8767adce28346c23f9c5fe2934e136864bc3a6d9731","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302080,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[1,-6]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[1, -6]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0081","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d36a806c8da1177a577ec8d5446df2d7566afb780923d458c0f64e631d74fca","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302081,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[7,2],[2,5]],\"x\":[1,-2],\"y\":[2,-7]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[2, -7]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0082","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df16ae3a6a1e1998612a752747778c9ee79c6df25ccc0d12d1cdf9ca48a6239f","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302082,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[8,0],[0,1]],\"x\":[1,-1],\"y\":[0,-8]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[0, -8]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0083","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e25fee78a8694e08bdc1568f3a9a46c9478c46e2afdea50d63485a7480716a22","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302083,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[9,1],[1,2]],\"x\":[1,0],\"y\":[1,-9]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[1, -9]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 0], [1, -9], 0, 162, 162, Yes)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [1, -9], 0, 162, 162, Yes)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0084","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5592147500d16e8a0c78f695b7ae9221745084c23746c3576786949db109f3af","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302084,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[10,2],[2,5]],\"x\":[1,1],\"y\":[2,-10]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[2, -10]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0085","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92333ecb0d25d4e5cd43e1985c251b94cfce30b1fa1716167b57de067e73f525","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302085,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[11,0],[0,1]],\"x\":[1,2],\"y\":[0,-11]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[0, -11]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0086","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d0eae231c8e71a142a586e1ad9a2aaec7882f62105297620c555dbe9e051ff","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302086,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[12,1],[1,2]],\"x\":[1,-2],\"y\":[1,-12]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[1, -12]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0087","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0539ef4ee06ad3fafbc8cb502a8570f6d1f58d22c835e6009679130c2c168a","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302087,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[2,-13]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[2, -13]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0088","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25187a01aa6e41f49268e1daaac3cf5e92ceedb5687b5c0a57d9358d091ed622","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302088,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[14,0],[0,1]],\"x\":[1,0],\"y\":[0,-14]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[0, -14]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 0], [0, -14], 0, 210, 210, Yes)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [0, -14], 0, 210, 210, Yes)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0089","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce3849fbb92a6fd3e4a2ee607ad7213f99ff6d72bfc2fa8887539fa3f26529e","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302089,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[15,1],[1,2]],\"x\":[1,1],\"y\":[1,-15]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[1, -15]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0090","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"449e2ec26b4f20f92fdfec792d284031e2989a1e8ac12cd2735c4e9f4e09e5ac","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302090,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[2, -16]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0091","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c17056d15df7aab583ca9cbe71282935f992c2b36d806d1a668deb8e89d2a1f4","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302091,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[17,0],[0,1]],\"x\":[1,-2],\"y\":[0,-17]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[0, -17]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0092","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a0565d706b431bf12dc28b5fc3afbd8e774bf425af949e38f51373e785c3301","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302092,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[18,1],[1,2]],\"x\":[1,-1],\"y\":[1,-18]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[1, -18]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0093","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0fa0ea0a108648338026c8d693848596bd56cb07776b406f81434b3a51c05d4","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302093,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[19,2],[2,5]],\"x\":[1,0],\"y\":[2,-19]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[2, -19]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 0], [2, -19], 0, 1748, 1748, Yes)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [2, -19], 0, 1748, 1748, Yes)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0094","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"751cf80566c5b66ccdd3be30476a08dc0ccf0c4d8cf8bf124213be05369a1fe0","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302094,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[0,-20]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[0, -20]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0095","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b4ed5fd8a7501b0bd0a57d29428fb3d7d578096a58a4e93fa3713577174da8","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302095,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[21,1],[1,2]],\"x\":[1,2],\"y\":[1,-21]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[1, -21]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0096","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa76cce77b1b855ad475d1c6e74f48a075314f3d1d5ea7e1f007c2e084da30e","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302096,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[22,2],[2,5]],\"x\":[1,-2],\"y\":[2,-22]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[2, -22]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0097","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dacaae055c935b534deb9cddd4a23a715cac64070310931810088df627a91a05","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302097,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[23,0],[0,1]],\"x\":[1,-1],\"y\":[0,-23]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[0, -23]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0098","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b05b693dbf50057cc0daf34130553d85283d908855ece777b1e4be11986d41bf","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302098,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[24,1],[1,2]],\"x\":[1,0],\"y\":[1,-24]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[1, -24]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 0], [1, -24], 0, 1152, 1152, Yes)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [1, -24], 0, 1152, 1152, Yes)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0099","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629e6f09916aae8c20859c3e139855233a5b0ef0c2b1d90d28c2abdc3c65676b","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302099,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[25,2],[2,5]],\"x\":[1,1],\"y\":[2,-25]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[2, -25]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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-20261030-r02-real-hilbert-space-0100","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efb759542eadce85ec6f3fb905787960760196b1b8358cd7ef95886abc022473","family_id":"qf-mod30-hilbert-pythagorean","seed":202610302100,"semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[26,0],[0,1]],\"x\":[1,2],\"y\":[0,-26]}"},"fields":{"title":"Check Orthogonality before Using the Inner-Product Pythagorean Identity","stem":"Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[0, -26]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).","answer":{"kind":"expression","value":"([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false)","explanation":"Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false)."},"skills":["Check Orthogonality before Using the Inner-Product Pythagorean Identity"],"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."}}