{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r03-real-banach-space-0001","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"031f49817e18cd87ee45ba3610dd0bb4bfce1bb4ed5c283a75c0a10f2094f410","family_id":"qf-mod28-banach-named-certificate","seed":202610283001,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":1,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=1, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(1, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0002","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e35a0833b9a3440be6561f249706200cb2e6f344d35579edc0fb27c15e0d288b","family_id":"qf-mod28-banach-named-certificate","seed":202610283002,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":2,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=2, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(2, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0003","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57e7785cdf7c9a0fd6b5cf5bc4b2d94e2f73f18c946badf74a09850fe31ec53e","family_id":"qf-mod28-banach-named-certificate","seed":202610283003,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":3,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(3, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0004","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85905bdfbaea32c08f062dae60755a26589bbfc11534be3c9d797e6b61383eef","family_id":"qf-mod28-banach-named-certificate","seed":202610283004,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":4,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=4, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(4, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0005","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"849bc201c239574cd9dc94db4964a66c9c497fbc056afc383d58d5bab5da043b","family_id":"qf-mod28-banach-named-certificate","seed":202610283005,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":5,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=5, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(5, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0006","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6c4bdb8a0840ed19dc4d5608117d3c0999ea0c9d7c74eac7a2db0663bf30b20","family_id":"qf-mod28-banach-named-certificate","seed":202610283006,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":6,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=6, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(6, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0007","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b9029ba0ac97c67f5457b5e17d26b434a5f165ab51bbffc791483b798fa01f8","family_id":"qf-mod28-banach-named-certificate","seed":202610283007,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":7,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=7, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(7, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (7, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0008","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80971271b1d56ed04541659ac8ac1fd1a610a6a22614157c9a5109749a54bff5","family_id":"qf-mod28-banach-named-certificate","seed":202610283008,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":8,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=8, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(8, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (8, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0009","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eafb4d95eed5420ddabb1061776bcf5d831dc63e9adf7853083454704297c5f","family_id":"qf-mod28-banach-named-certificate","seed":202610283009,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":9,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=9, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(9, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (9, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0010","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb1b81129a6d27146dd06e359c4c0baf25ac3661f10720d2a7a3f3d74bbd190","family_id":"qf-mod28-banach-named-certificate","seed":202610283010,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":10,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=10, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(10, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (10, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0011","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddf57dfb504fb0b1ad3d677e854014fd620fc8e791f587142baf608e5827beec","family_id":"qf-mod28-banach-named-certificate","seed":202610283011,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":11,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=11, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(11, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (11, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0012","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a41cf8825e653f97f41531ac652c6a4bbaf4212fd258bb943e4a7203d75db18","family_id":"qf-mod28-banach-named-certificate","seed":202610283012,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":12,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=12, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0013","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c5e1fcb9f108cb64f84e27d4cec8f878e4b5f248e7cdda5d813cff250057bc8","family_id":"qf-mod28-banach-named-certificate","seed":202610283013,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":13,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=13, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(13, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (13, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0014","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b6539d22d54a70d00c7fafc7878fa912a56c21a519875437b032352026586bb","family_id":"qf-mod28-banach-named-certificate","seed":202610283014,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":14,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=14, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(14, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (14, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0015","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d086999c6ae0ac8eb8fb0104eb36af563a0120df0fb5e9d52af386ed82df6c76","family_id":"qf-mod28-banach-named-certificate","seed":202610283015,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":15,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=15, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(15, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (15, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0016","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15a12bc649a5349f688765678845213841a3a4c58a415621f3caad889286d47","family_id":"qf-mod28-banach-named-certificate","seed":202610283016,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":16,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=16, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(16, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (16, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0017","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3342103e357c3c99e122088d5aca53fa5c8e02cf74f73b5a91d0b2912f05d37c","family_id":"qf-mod28-banach-named-certificate","seed":202610283017,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":17,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=17, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(17, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (17, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0018","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9e25ec1dc9df13cdc712a5b4be3c1edf76e68126107265c4c9945f867caa26","family_id":"qf-mod28-banach-named-certificate","seed":202610283018,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":18,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=18, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(18, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (18, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0019","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"388da0f83771c6585452605ecf1c287ee5b5cc50915ba2d79e0ec6e592a8c39a","family_id":"qf-mod28-banach-named-certificate","seed":202610283019,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":19,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=19, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(19, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (19, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0020","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f83c9581f92b3f3b83f3496fcb1a0f21bf2dfe4a1ed4f116e6983977401e501","family_id":"qf-mod28-banach-named-certificate","seed":202610283020,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":20,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=20, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(20, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (20, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0021","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59c00bbf03bdde54a488439b0aecd577dc07a7466438df494df5d98621b50456","family_id":"qf-mod28-banach-named-certificate","seed":202610283021,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":21,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=21, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(21, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (21, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0022","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56e4258ce788da85ff20d6314be8b0aeeb7093d53699e83d66ec3c2d4673ff6d","family_id":"qf-mod28-banach-named-certificate","seed":202610283022,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":22,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=22, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(22, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (22, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0023","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91089e0cba94e129314f2e8e94473f3bd55cbfb6accc317c8331f893c4d4b925","family_id":"qf-mod28-banach-named-certificate","seed":202610283023,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":23,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=23, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(23, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (23, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0024","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e7d606b3493b0eaec051ccc1fba6b1533e21a4ad8bcead9b01b9d51901aed0","family_id":"qf-mod28-banach-named-certificate","seed":202610283024,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":24,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=24, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(24, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (24, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0025","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2e13f2e29aa5c0d17dd29ec76e5c5bc64bfa65ad465332ef161ef04d1eb0a07","family_id":"qf-mod28-banach-named-certificate","seed":202610283025,"semantic_template":"qf-mod28-banach-named-certificate/{\"dimension\":25,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named Finite-Dimensional Banach Certificate Binds One Space","stem":"Use the explicit real coordinate space R^n of dimension=25, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).","answer":{"kind":"expression","value":"(25, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)","explanation":"Invoke the named completeness theorem for this exact real coordinate space with its stated norm-induced metric. Completeness in an unrelated metric would not supply this certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (25, l1, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)."},"skills":["A Named Finite-Dimensional Banach Certificate Binds One 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-20261028-r03-real-banach-space-0026","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee711fa0a45fae4cc84b2141191a7d116c81628a962207cb25dd1398c35e8e0","family_id":"qf-mod28-banach-component-checklist","seed":202610283026,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":1,\"metric_source\":\"unrelated_metric\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=1, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(1, l1, unrelated_metric, sampled_sequences, No, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, l1, unrelated_metric, sampled_sequences, No, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0027","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7272cb89e06b493d9c51bb2af568ef90c44ce91d465b0957c977c7e8f41a9c39","family_id":"qf-mod28-banach-component-checklist","seed":202610283027,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=2, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(2, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0028","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"710183ac4da687d5447820f67973fd09df1e1d5874a08dfdde6717aa18c1e3e6","family_id":"qf-mod28-banach-component-checklist","seed":202610283028,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":3,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(3, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0029","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84088f6f60f155c1fb5db0a2593321d383eb371649d2820fdfb70ba2a651ed4b","family_id":"qf-mod28-banach-component-checklist","seed":202610283029,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":4,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=4, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(4, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0030","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59c497d9ded57d2aaa0931ed3645078fd3a813947d861ba90b71936bb7c97aa1","family_id":"qf-mod28-banach-component-checklist","seed":202610283030,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":5,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=5, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(5, l1, same_norm_induced, sampled_sequences, Yes, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, l1, same_norm_induced, sampled_sequences, Yes, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0031","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9dc2d1c4f48005d34070f51aec646de58454598f8283a335ed0f22cc80b9d46","family_id":"qf-mod28-banach-component-checklist","seed":202610283031,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":6,\"metric_source\":\"unrelated_metric\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=6, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(6, linfinity, unrelated_metric, finite_dimensional_theorem, No, Yes, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, linfinity, unrelated_metric, finite_dimensional_theorem, No, Yes, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0032","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33bcbc002f6067edeef757b6b5d69fedaf511705b9560ce3621cd661ba4bc283","family_id":"qf-mod28-banach-component-checklist","seed":202610283032,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":7,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=7, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(7, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (7, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0033","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f10e03ab28044641937aa9244890ecbf39e48faaf1a7ca9c9ef17789e05bfe80","family_id":"qf-mod28-banach-component-checklist","seed":202610283033,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":8,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=8, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(8, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (8, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0034","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a0a82c501959bad6a0e576ab373602b3a56e54dd12e742458b635157208b6b9","family_id":"qf-mod28-banach-component-checklist","seed":202610283034,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":9,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=9, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(9, l1, same_norm_induced, sampled_sequences, Yes, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (9, l1, same_norm_induced, sampled_sequences, Yes, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0035","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aad2901390daf97cf77198bef29c6fce07351082ae8d31bc14ed77574bb1599","family_id":"qf-mod28-banach-component-checklist","seed":202610283035,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":10,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=10, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(10, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (10, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0036","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d84a5bdec97b32eed8aa7758d8af211209c7c96861957d14e7d20f365ef5c85","family_id":"qf-mod28-banach-component-checklist","seed":202610283036,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":11,\"metric_source\":\"unrelated_metric\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=11, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(11, l1, unrelated_metric, finite_dimensional_theorem, No, Yes, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (11, l1, unrelated_metric, finite_dimensional_theorem, No, Yes, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0037","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72cd5675fcc405aa691fdafca198468cf17472aa439b8c426b5f4f96144595ac","family_id":"qf-mod28-banach-component-checklist","seed":202610283037,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":12,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=12, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(12, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0038","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cf5bcfd5991aec0f4293a7da045a9014f5572f46fd1213d65be3361e5e31f57","family_id":"qf-mod28-banach-component-checklist","seed":202610283038,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":13,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=13, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(13, l1, same_norm_induced, sampled_sequences, Yes, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (13, l1, same_norm_induced, sampled_sequences, Yes, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0039","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83bb5166dab1bf673c99055657465bfa622f3af8be1c69aac38b81b42c7bf60b","family_id":"qf-mod28-banach-component-checklist","seed":202610283039,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":14,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=14, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(14, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (14, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0040","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019c0001dab33b82896ec57d88ebbb1e0f8dbda2a112dd8834064fa57316d022","family_id":"qf-mod28-banach-component-checklist","seed":202610283040,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":15,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=15, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(15, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (15, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0041","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3644199f5415b7b75dede58f66cb94d8544b47b723ce8fe76e2fb3f124c45546","family_id":"qf-mod28-banach-component-checklist","seed":202610283041,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":16,\"metric_source\":\"unrelated_metric\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=16, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(16, linfinity, unrelated_metric, finite_dimensional_theorem, No, Yes, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (16, linfinity, unrelated_metric, finite_dimensional_theorem, No, Yes, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0042","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e82f7d370f337967373d79e3beb29c82fad0209c88388187f9ddc4ddcdceba0","family_id":"qf-mod28-banach-component-checklist","seed":202610283042,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":17,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=17, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(17, l1, same_norm_induced, sampled_sequences, Yes, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (17, l1, same_norm_induced, sampled_sequences, Yes, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0043","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"772382074b4b85c51e6bfd5b735548bd9056dc4f52902f9b17c7de175be52541","family_id":"qf-mod28-banach-component-checklist","seed":202610283043,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":18,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=18, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(18, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (18, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0044","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13f0a212ad5b351f9a3d3eddac9a80ddd6390a12e21fb1b2b41c7217dd1af671","family_id":"qf-mod28-banach-component-checklist","seed":202610283044,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":19,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=19, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(19, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (19, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0045","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7469e3320ae7d120f0f8134be3dee7d5140d9c3842108755f4547abb5cc59017","family_id":"qf-mod28-banach-component-checklist","seed":202610283045,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":20,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=20, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(20, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (20, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0046","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f47c355b2b8eae9f88fc833f28c11fe662740695e88c8921a9fa7654038aa888","family_id":"qf-mod28-banach-component-checklist","seed":202610283046,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":21,\"metric_source\":\"unrelated_metric\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=21, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(21, l1, unrelated_metric, sampled_sequences, No, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (21, l1, unrelated_metric, sampled_sequences, No, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0047","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"891929115c029b3076dfd24bf91cd9008b935044433ab58102aa817e51a8dc25","family_id":"qf-mod28-banach-component-checklist","seed":202610283047,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":22,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=22, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(22, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (22, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0048","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a53b6b70d4ea9a4fb2e2ff1cf320809405d26d8853c57de5dba9c5d5556dde9","family_id":"qf-mod28-banach-component-checklist","seed":202610283048,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":23,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=23, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(23, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (23, l1, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0049","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5767047e0953f2f9a365d3cf02ce9430900fefb80fb4d54b51ba4ddfd219830","family_id":"qf-mod28-banach-component-checklist","seed":202610283049,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":24,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=24, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(24, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (24, linfinity, same_norm_induced, finite_dimensional_theorem, Yes, Yes, Yes)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0050","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e149c115a22c8606601bf6debc2ed8fddccb2be9975e793c9030051953fd98","family_id":"qf-mod28-banach-component-checklist","seed":202610283050,"semantic_template":"qf-mod28-banach-component-checklist/{\"completeness_source\":\"sampled_sequences\",\"dimension\":25,\"metric_source\":\"same_norm_induced\",\"norm_name\":\"l1\"}"},"fields":{"title":"Keep the Metric Binding and Completeness Source Separate","stem":"Use the explicit real coordinate space R^n of dimension=25, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=same_norm_induced, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.","answer":{"kind":"expression","value":"(25, l1, same_norm_induced, sampled_sequences, Yes, No, No)","explanation":"The checklist flags record exact matches to those declared certificate sources. Their conjunction qualifies this record; a missing binding does not refute the general finite-dimensional completeness theorem. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (25, l1, same_norm_induced, sampled_sequences, Yes, No, No)."},"skills":["Keep the Metric Binding and Completeness Source Separate"],"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-20261028-r03-real-banach-space-0051","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49cba752e8c53819cd1203f777bb8d4a1bc4c8dcaf49ee2a0ad33c127fc59661","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283051,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-12,-3,-5],\"norm_name\":\"l1\",\"right\":[12,-6,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-12, -3, -5], right=[12, -6, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0052","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd43873c65b3bbe2a9cdfc05c37cacd2881fc6698fc2264a605e4b48275833bf","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283052,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-11,-2,-2],\"norm_name\":\"linfinity\",\"right\":[11,-1,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-11, -2, -2], right=[11, -1, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0053","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb8b539f73b33a69ec8f63e98dba0e2d16c612d6a9576729c65800472e00a4a3","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283053,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-10,-1,1],\"norm_name\":\"l1\",\"right\":[10,4,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-10, -1, 1], right=[10, 4, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0054","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"886538bdbeb5f2c2de1f5805e29e626ebabe0a33d800372f209ccc914f8a44aa","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283054,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-9,0,4],\"norm_name\":\"linfinity\",\"right\":[9,-4,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-9, 0, 4], right=[9, -4, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0055","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0f2c06e9d0ba895ece9492902aef877aa1cad6193218fff8ce6bfa6f6c90c0f","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283055,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-8,1,-4],\"norm_name\":\"l1\",\"right\":[8,1,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-8, 1, -4], right=[8, 1, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0056","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ce22c0a62fe95c68e7bedea9515a598dcf9f8fe61baaad937cf15e03971c3be","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283056,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-7,2,-1],\"norm_name\":\"linfinity\",\"right\":[7,6,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-7, 2, -1], right=[7, 6, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0057","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14940da2f722705e8e661de9d6866c42a222b9ac07eb6ce5cc34ca5cc7eb61e2","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283057,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-6,3,2],\"norm_name\":\"l1\",\"right\":[6,-2,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-6, 3, 2], right=[6, -2, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0058","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65476abfb0389772cca968887ded8e44008acbf03af635b49d61d7cc3abe2d1b","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283058,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-5,-3,5],\"norm_name\":\"linfinity\",\"right\":[5,3,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-5, -3, 5], right=[5, 3, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0059","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f599bf981e7b675830b40411e7d9ea0410606410ff646acc0f8acb7d7e4a0d45","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283059,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-4,-2,-3],\"norm_name\":\"l1\",\"right\":[4,-5,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0060","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18b6f65a0107ee80034fbc3e7519c7afe0bfeef650ec3ded4f4ba3330800f7aa","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283060,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-3,-1,0],\"norm_name\":\"linfinity\",\"right\":[3,0,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-3, -1, 0], right=[3, 0, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0061","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e937ceb71e7ef4cf78744bfce49107aa59ce3b01116849815249c774d42b2b2","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283061,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-2,0,3],\"norm_name\":\"l1\",\"right\":[2,5,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-2, 0, 3], right=[2, 5, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0062","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a42980c27d63f3705bb1a6350030e9711e1fc5875691c121e43bcb56f3a46361","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283062,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[-1,1,-5],\"norm_name\":\"linfinity\",\"right\":[1,-3,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[-1, 1, -5], right=[1, -3, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0063","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba8d9f42ca47168cbe115fccb80ef3494ac8406b888671e9e8798467ed59afa","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283063,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[0,2,-2],\"norm_name\":\"l1\",\"right\":[0,2,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0064","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aef509927ab6ac3195c2ef258046a6f7cb1f2853830388beccecd5db1bfa6138","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283064,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[1,3,1],\"norm_name\":\"linfinity\",\"right\":[-1,-6,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[1, 3, 1], right=[-1, -6, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0065","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c6546ee164048c66fb315d990a7abe16dcf10cd380fb55a606f4b121a4e82ab","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283065,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[2,-3,4],\"norm_name\":\"l1\",\"right\":[-2,-1,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0066","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"689c8fbd24bbbdedbf36552c9d21f21de5f6b2cfe924952207719aa54480d66b","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283066,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[3,-2,-4],\"norm_name\":\"linfinity\",\"right\":[-3,4,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[3, -2, -4], right=[-3, 4, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0067","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76f8092c9288f38daffc9bc358d014f3430418bb151d391c4c3abde23a9c9b68","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283067,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[4,-1,-1],\"norm_name\":\"l1\",\"right\":[-4,-4,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[4, -1, -1], right=[-4, -4, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0068","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df8ed02de4fc32c8e04280babe74ed1acc152e4ee51b4d5b5eb50044d4b51170","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283068,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[5, 0, 2], right=[-5, 1, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0069","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e82aa78fb80e6a3394062503033cb8e6df8bcd0f0536c7d0b86c723725f493","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283069,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[6,1,5],\"norm_name\":\"l1\",\"right\":[-6,6,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[6, 1, 5], right=[-6, 6, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0070","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0dbe8b09625796448ef577fc6c4faffda63226ec4777abe6fd64e2eae349df3","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283070,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[7,2,-3],\"norm_name\":\"linfinity\",\"right\":[-7,-2,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[7, 2, -3], right=[-7, -2, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0071","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"916346284bacb33d13f4f3448617884644226c3cc1c5439c075b482c129fa942","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283071,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[8,3,0],\"norm_name\":\"l1\",\"right\":[-8,3,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0072","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9107725f5b7ab6e6b6fa69fc79cd9eebff333ef60e4e42d37df4e19c87aa497a","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283072,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[9,-3,3],\"norm_name\":\"linfinity\",\"right\":[-9,-5,-1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[9, -3, 3], right=[-9, -5, -1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0073","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe875e778367e439dfcbfe3a86ee4371a872880e2cb9ace1219cf35be9b9391","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283073,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[10,-2,-5],\"norm_name\":\"l1\",\"right\":[-10,0,0]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[10, -2, -5], right=[-10, 0, 0]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0074","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95573de5d76af91fc6e565f730b6641882d63db8d14ebf5b0e3d638e283f119b","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283074,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0075","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a088b492960aca7294c374e1798187fd389a9b1f09c8a759b78f6ae190500d1e","family_id":"qf-mod28-banach-distance-evaluate","seed":202610283075,"semantic_template":"qf-mod28-banach-distance-evaluate/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"},"fields":{"title":"A Named-Norm Distance Is Not a Completeness Proof","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).","answer":{"kind":"expression","value":"(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)","explanation":"Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)."},"skills":["A Named-Norm Distance Is Not a Completeness Proof"],"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-20261028-r03-real-banach-space-0076","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811b7d7ee172b96bdd4d372e26fd5035a3962ba83602d6c6f392f708e443426c","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283076,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":1,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=1, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(1, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0077","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8319541386ba5c9d7b92e0b51b1688b8186c2f3deadbbdf3554d0c5e98cf83b","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283077,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":2,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=2, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(2, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0078","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2072bfd6a7e2465584b09488d074d4d6727057d7fee28d93326cd52a051f334","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283078,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":3,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(3, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0079","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec35128ca4d939e6f5aa2ab0266e00e61e60848e8ea534d5110a6889152916c","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283079,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":4,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=4, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(4, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0080","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f22f40187d7d0c82f47a5cdffa2321ee73f900d207a0617b6ecd329b2a44194","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283080,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":5,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=5, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(5, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0081","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05e75a43a1ad73ca0e1df3a83515394ba1831acaf716435a7036ad5ba0eb56f2","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283081,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":6,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=6, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(6, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0082","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a67333a0b1cf38a711dc3b8d0459b8092f1f6d0fcb723ed2d8fe0a06b3cb730f","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283082,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":7,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=7, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(7, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (7, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0083","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd50b561208755f63a5cf523c684d8728bc9deb37ce38ec27c609a1efb45059","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283083,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":8,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=8, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(8, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (8, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0084","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d165cf152a7bd35a868c2ffa86b2d5c2e1d1bc4af697a4fd5c631437982d1a13","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283084,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":9,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=9, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(9, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (9, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0085","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebf9d7f1097a6b8c91340b7d7293282082a76c2b98ec502bc1469c28120ecd29","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283085,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":10,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=10, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(10, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (10, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0086","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f9f5daada43ab0bf19ed64aefbecc92b8afdebe85afc9819d7ae9a8e9e3e418","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283086,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":11,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=11, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(11, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (11, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0087","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86210f67e74783e133ae7aa4ee993a293175d1eb08de99cdfbd8bf39bc2d1135","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283087,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":12,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=12, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(12, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (12, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0088","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09adbbe558a7a9e4ffa3fd516fdcee468423692f0bb0c2861edf4ed3a03e79d1","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283088,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":13,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=13, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(13, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (13, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0089","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c33dc394d77e3a2a1b27a8453866a8640f16779483d80ca41395e55fe3020683","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283089,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":14,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=14, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(14, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (14, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0090","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ce2f9318693618b34787b46bfa7c2360ee4d1b47e7c3f0faff58c64fadd8c2","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283090,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":15,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=15, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(15, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (15, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0091","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2598d9097e04fd7b0654affef5b9d39bb7b6099d86c852a7c75a0c381e98c091","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283091,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":16,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=16, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(16, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (16, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0092","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0367b5302a79cafa0bcbfc3f1d5c8a6dfb79325c0815b0d572ee0cab838a8827","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283092,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":17,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=17, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(17, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (17, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0093","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d80ca2f529cea4f98113ecdfc66741b8a0d46a9438c42a19045c6ebb9c5f6779","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283093,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":18,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=18, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(18, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (18, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0094","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44999c1be39f879555471b85c7e64cc6e767f7f03765725cd8d8558eafdb6c09","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283094,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":19,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=19, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(19, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (19, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0095","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffdd4657f8f9f88cde6ce0bdff819744a03ae4a729d5e1586454d5b4679dece2","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283095,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":20,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=20, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(20, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (20, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0096","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cb39d8a9d319e72f320e71d8b321773729c6c561cb69cdbeef7e0cee8df62b5","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283096,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":21,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=21, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(21, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (21, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0097","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4873689dff7457a6f6c7cee11d963cce519b6395184b7e4319dc158b984afcf1","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283097,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":22,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=22, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(22, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (22, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0098","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bb8596d0dbb1854958de3ad647efda30a974e98f03d34fd39d9787dfbe79ad5","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283098,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":23,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=23, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(23, l1, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (23, l1, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0099","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7df624d08950b1845c0432f18063f641002e5e02f4cbb19bd4c9ab880ba15454","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283099,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":24,\"evidence_mode\":\"finite_dimensional_theorem\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=24, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(24, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (24, linfinity, finite_dimensional_theorem, Yes, universal_named_certificate)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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-20261028-r03-real-banach-space-0100","batch_id":"batch-20261014-advance-20261028-r03-real-banach-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"572a4c0e296505957ecf8e200d22969488e508098069a872ee0eb5955534e159","family_id":"qf-mod28-banach-evidence-mode-audit","seed":202610283100,"semantic_template":"qf-mod28-banach-evidence-mode-audit/{\"dimension\":25,\"evidence_mode\":\"sampled_cauchy_sequences\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Completeness Theorem from Sampled Cauchy Sequences","stem":"Use the explicit real coordinate space R^n of dimension=25, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.","answer":{"kind":"expression","value":"(25, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)","explanation":"The sample-mode No is a statement about insufficient evidence, not a classification of this named space as incomplete. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (25, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)."},"skills":["Distinguish a Completeness Theorem from Sampled Cauchy Sequences"],"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."}}