{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0001","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60922d4e28d40e14613c6d6d70669a3d6221e01ffb7173e90dd55254766f9208","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262001,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-12,-8,-9,-2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-12, -8, -9, -2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-12, -8, -9, -2], l1, 31)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -8, -9, -2], l1, 31)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0002","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b914f98d9c6789b90eb20be41b99ac40dee270e3b28cd93bb222b6a7ad2db606","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262002,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-11,-7,-2,-1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-11, -7, -2, -1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-11, -7, -2, -1], l1, 21)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -7, -2, -1], l1, 21)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0003","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"896abd6a598b5022125037013dbd068508fcaf7ebb194a84687d6cb1b01f2091","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262003,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-10,-4,5,0]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-10, -4, 5, 0]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-10, -4, 5, 0], l1, 19)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -4, 5, 0], l1, 19)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0004","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f046b8395b2aa37f6ad7d68d047d2a29d51a7beaa18f54bdf07dcb03881a9ac9","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262004,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-9,1,-7,1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-9, 1, -7, 1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-9, 1, -7, 1], l1, 18)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 1, -7, 1], l1, 18)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0005","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091db1d9f8dcf4dc6cf1002acd519d5d6d07f28ce7d76b06f1bc959e59a2b6b1","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262005,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-8,8,0,2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-8, 8, 0, 2], l1, 18)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 8, 0, 2], l1, 18)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0006","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2012f895efaaedda267b8173637dd0512f58b4f867d01e0ec15d6d6d6542b70","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262006,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-7,0,7,-2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-7, 0, 7, -2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-7, 0, 7, -2], l1, 16)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 0, 7, -2], l1, 16)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0007","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c91149e144deedc8f97b60fc1b142203d28cfcd10409114ee2ed0c075b59df4c","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262007,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-6,-6,-5,-1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-6, -6, -5, -1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-6, -6, -5, -1], l1, 18)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -6, -5, -1], l1, 18)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0008","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e76944c44ee2a838a1ec9b27d4e3e92cfdf6822367d066279f38ae3bdea7e2dc","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262008,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-5,7,2,0]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-5, 7, 2, 0]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-5, 7, 2, 0], l1, 14)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 7, 2, 0], l1, 14)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0009","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e7a242c92c1a2dd6dfd9e7558d5917c440b53979c02c6b1e6773f2c0aa59b05","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262009,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-4,5,9,1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-4, 5, 9, 1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-4, 5, 9, 1], l1, 19)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 5, 9, 1], l1, 19)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0010","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb96687b8d8c50b2efd9216f74558466b4b08faca076693fe8d6e4375fb0d3c4","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262010,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-3,5,-3,2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-3, 5, -3, 2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-3, 5, -3, 2], l1, 13)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -3, 2], l1, 13)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0011","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"368deb4320cbe7a49f3ee9e5d1dcde52d7dd073adc0be85cf05aca347f7b09c6","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262011,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-2,7,4,-2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-2, 7, 4, -2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-2, 7, 4, -2], l1, 15)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 7, 4, -2], l1, 15)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0012","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb127aedcf0a35749322f9072dd97b52c375bd4d344069a4c46c98cd655338a","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262012,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[-1,-6,-8,-1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-1, -6, -8, -1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-1, -6, -8, -1], l1, 16)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, -6, -8, -1], l1, 16)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0013","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d73e9415423dff520d88a25e207b3e17a003d78e399af5ce718e6668b82d5fa4","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262013,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[0,0,-1,0]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[0, 0, -1, 0]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([0, 0, -1, 0], l1, 1)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 0, -1, 0], l1, 1)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0014","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6afc12b6a835bb3c6532e73ead386a7cce834b5aa3469ae1d5c5dbdfaf2bf4","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262014,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[1,8,6,1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[1, 8, 6, 1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([1, 8, 6, 1], l1, 16)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 8, 6, 1], l1, 16)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0015","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17bcdcfbc6b2090d33ec95c34642f87da29df0c6a2703718c54730f4bdd36887","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262015,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[2,1,-6,2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[2, 1, -6, 2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([2, 1, -6, 2], l1, 11)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, -6, 2], l1, 11)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0016","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8cdeb75667231c176718f381958f2d70af6efe41d70f1521cb8fd483301d24e","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262016,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[3,-4,1,-2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[3, -4, 1, -2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([3, -4, 1, -2], l1, 10)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, -4, 1, -2], l1, 10)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0017","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd297c29b489b2cecb29475398aea2eb369e392429e11f80c09ed28dee02a2e8","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262017,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[4,-7,8,-1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[4, -7, 8, -1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([4, -7, 8, -1], l1, 20)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -7, 8, -1], l1, 20)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0018","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8509644bae66e8f666a8c51f13fcb852bb7ab50db44213acc81795b72680b153","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262018,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[5,-8,-4,0]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[5, -8, -4, 0]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([5, -8, -4, 0], l1, 17)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, -8, -4, 0], l1, 17)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0019","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040a499f34cfe3a134764a7729cb962a280b0c2bfedd720801751258acbf8c37","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262019,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[6,-7,3,1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[6, -7, 3, 1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([6, -7, 3, 1], l1, 17)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, -7, 3, 1], l1, 17)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0020","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36bdccae465fd8229700d6fec449582fc59760f653f6e6bddf00d4619ef2a1ed","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262020,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[7,-4,-9,2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[7, -4, -9, 2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([7, -4, -9, 2], l1, 22)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, -4, -9, 2], l1, 22)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0021","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"686828e124564f69550b7fe1477c9e26009ccf8b3d5dc266911b124db9805d1c","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262021,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[8,1,-2,-2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[8, 1, -2, -2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([8, 1, -2, -2], l1, 13)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1, -2, -2], l1, 13)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0022","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38eaf18362fb82aec8962804b05c23d24a1878280d0f536baf390f85a374d449","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262022,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[9,8,5,-1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[9, 8, 5, -1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([9, 8, 5, -1], l1, 23)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 8, 5, -1], l1, 23)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0023","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4028cdd949f225ef39e4de8b68360ce24d4c7d3be189a3749e85539e39fe6e57","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262023,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[10,0,-7,0]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[10, 0, -7, 0]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([10, 0, -7, 0], l1, 17)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 0, -7, 0], l1, 17)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0024","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e389168ff6e63e11dc4ce71014040c7b1fa89f11d5cf9533d6d6b5375c088073","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262024,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[11,-6,0,1]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[11, -6, 0, 1]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([11, -6, 0, 1], l1, 18)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, -6, 0, 1], l1, 18)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0025","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf46b3afa872d58fdd91bf4702be77c3123b4ec2ec0db60eb49dbfb598aba968","family_id":"qf-mod26-norm-l1-evaluate","seed":202610262025,"semantic_template":"qf-mod26-norm-l1-evaluate/{\"vector\":[12,7,7,2]}"},"fields":{"title":"Evaluate a Named l1 Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[12, 7, 7, 2]. Evaluate l1(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([12, 7, 7, 2], l1, 28)","explanation":"Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 7, 7, 2], l1, 28)."},"skills":["Evaluate a Named l1 Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0026","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ed8e6f524fa0906352967ffbf590f9c02719f68453474ca094b0cbc7a092ffb","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262026,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-12,-8,-9,-2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-12, -8, -9, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-12, -8, -9, -2], linfinity, 12)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -8, -9, -2], linfinity, 12)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0027","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47e6aa1070a43b7ceecb03dbaf21d7f92237de5de6599777875c3500ba6ab377","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262027,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-11,-7,-2,-1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-11, -7, -2, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-11, -7, -2, -1], linfinity, 11)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -7, -2, -1], linfinity, 11)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0028","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fa9584a87230e404f1514a17704662e5925a8924cad86f03e4625796a3dbb88","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262028,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-10,-4,5,0]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-10, -4, 5, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-10, -4, 5, 0], linfinity, 10)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -4, 5, 0], linfinity, 10)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0029","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c195120458a48804467fb4eebc755730acc74f6cacf327dbacd2067d72855ee9","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262029,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-9,1,-7,1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-9, 1, -7, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-9, 1, -7, 1], linfinity, 9)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 1, -7, 1], linfinity, 9)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0030","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30185a25cee8394eb3a1a7189bdf544b5fc89f225690c81c599604c3893ec667","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262030,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-8,8,0,2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-8, 8, 0, 2], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 8, 0, 2], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0031","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"913945e405c9fa42f8b2c62580c2031520e9e3e42796548b64d09b34d489cf90","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262031,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-7,0,7,-2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-7, 0, 7, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-7, 0, 7, -2], linfinity, 7)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 0, 7, -2], linfinity, 7)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0032","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a6b7dfa49562372cb565c403c8454b30318ddb4676dbbd39c313cbc8ef0ae70","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262032,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-6,-6,-5,-1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-6, -6, -5, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-6, -6, -5, -1], linfinity, 6)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -6, -5, -1], linfinity, 6)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0033","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31f6f054722ae53e72586d2932fb3a56ee5cd954061250de51b93f89b1d646c1","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262033,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-5,7,2,0]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-5, 7, 2, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-5, 7, 2, 0], linfinity, 7)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 7, 2, 0], linfinity, 7)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0034","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d61d46c1a66cc7f7b167e9122695c675a93b740d732d1b445a1e840975f1c443","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262034,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-4,5,9,1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-4, 5, 9, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-4, 5, 9, 1], linfinity, 9)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 5, 9, 1], linfinity, 9)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0035","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a220ecf52e1e6168961a09e0aa6bc747c3e2b99e67bde2aa60ab960ff8bbf182","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262035,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-3,5,-3,2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-3, 5, -3, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-3, 5, -3, 2], linfinity, 5)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -3, 2], linfinity, 5)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0036","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db138f930b11314527aa32cb55113873585b990f9c19a1cec400ff70f1547ea","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262036,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-2,7,4,-2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-2, 7, 4, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-2, 7, 4, -2], linfinity, 7)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 7, 4, -2], linfinity, 7)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0037","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfa40abf9eefb4c2f400e8c0a38d0605f4b27e1b65d7254659d663463f7809bb","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262037,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[-1,-6,-8,-1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-1, -6, -8, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([-1, -6, -8, -1], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, -6, -8, -1], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0038","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c45a925cc731448401ecd6560cbcf8319111070b9b4fcbf1597b13bc53c36ee","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262038,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[0,0,-1,0]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[0, 0, -1, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([0, 0, -1, 0], linfinity, 1)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 0, -1, 0], linfinity, 1)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0039","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec900435f6badbb70de6aea98b95b6f79c1afefd9d394ab797d6ec412264bc5","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262039,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[1,8,6,1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[1, 8, 6, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([1, 8, 6, 1], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 8, 6, 1], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0040","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"855e5a227e8b1123c08969aecbd53b76d4fade0bb8e473416f4f9465cdc7cd9b","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262040,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[2,1,-6,2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[2, 1, -6, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([2, 1, -6, 2], linfinity, 6)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, -6, 2], linfinity, 6)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0041","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"666582bf5f020978d501432b6437a51f139644ee4e487665508cd659e7281876","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262041,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[3,-4,1,-2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[3, -4, 1, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([3, -4, 1, -2], linfinity, 4)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, -4, 1, -2], linfinity, 4)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0042","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"313bed6f8a8a41da14b2018095717b200e87992dfec931b6c94a0759dffa4036","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262042,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[4,-7,8,-1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[4, -7, 8, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([4, -7, 8, -1], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -7, 8, -1], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0043","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dd7c5373b3b58385425a1269c33534a0ee05e401db8996c4c430c144f6dce93","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262043,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[5,-8,-4,0]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[5, -8, -4, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([5, -8, -4, 0], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, -8, -4, 0], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0044","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"900b192caddcbd9ca0ee3f51216e1fd15977e83728b05b3bd632be62beeb6960","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262044,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[6,-7,3,1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[6, -7, 3, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([6, -7, 3, 1], linfinity, 7)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, -7, 3, 1], linfinity, 7)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0045","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3421d8c488a4e0050e832446427ce213574792e555b0c5ca4f8ead9c6bc77a18","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262045,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[7,-4,-9,2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[7, -4, -9, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([7, -4, -9, 2], linfinity, 9)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, -4, -9, 2], linfinity, 9)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0046","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a45bc9bdd96e0855f070a4484088305bfe89c31825888b6716ccceaa3f9842e","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262046,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[8,1,-2,-2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[8, 1, -2, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([8, 1, -2, -2], linfinity, 8)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1, -2, -2], linfinity, 8)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0047","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f137d760ac5bde36e7d66761bc4d228cb7e093b04f5f2b0a49651beb442998fd","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262047,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[9,8,5,-1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[9, 8, 5, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([9, 8, 5, -1], linfinity, 9)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 8, 5, -1], linfinity, 9)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0048","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7de3a2a6e8af94e49500576326b8ee29da74d4aa3fc47e84056d8584b087ad9e","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262048,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[10,0,-7,0]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[10, 0, -7, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([10, 0, -7, 0], linfinity, 10)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 0, -7, 0], linfinity, 10)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0049","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f98540c010edf13d00298b527969bf0af2a6275063cd13d549396b17331dc55","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262049,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[11,-6,0,1]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[11, -6, 0, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([11, -6, 0, 1], linfinity, 11)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, -6, 0, 1], linfinity, 11)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0050","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff557dc9321a446a72e5bfd56d81c854ce87bf616fa1c277eb29e3741df0f1ec","family_id":"qf-mod26-norm-linfinity-evaluate","seed":202610262050,"semantic_template":"qf-mod26-norm-linfinity-evaluate/{\"vector\":[12,7,7,2]}"},"fields":{"title":"Evaluate a Named Maximum Norm Exactly","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[12, 7, 7, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).","answer":{"kind":"expression","value":"([12, 7, 7, 2], linfinity, 12)","explanation":"Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 7, 7, 2], linfinity, 12)."},"skills":["Evaluate a Named Maximum Norm Exactly"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0051","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f77c658bceee51305f759392bd0d45ef1c768ad191e45486b1ca2d3f59ce7b3","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262051,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":1,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=1. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 1, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 1, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0052","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b39086366def868cbf2ff7819bcea9e49cdbfa95eaf0b665ae12f35956ba3a7b","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262052,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":2,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=2. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 2, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0053","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84fee52bedf4abf1f974eb4771b7d870ad80c258faf0865c267a6f8764f5a93","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262053,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":3,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=3. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 3, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 3, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0054","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bdec95d29a8bc0d99446025c975f5296cf4d102e31a91fd2586a174ff18e3cf","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262054,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":4,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=4. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 4, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0055","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efd6bcdb2860e29d033f50071782e878ec4b7713ac793a94eb8de8c242da1351","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262055,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":5,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=5. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 5, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 5, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0056","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d04b6e14d80ca7ba86e6fa15395beb761ad0f3c98bf8c3187a4807fceb6a64e","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262056,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":6,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=6. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 6, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0057","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88ec403cd14fbbb72b2c1921476941774d30f6a4586016aabf5804a542e7da71","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262057,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":7,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=7. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 7, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0058","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71746888f5264f32ae2af50bde638166d0b57f4d6b9e5e26724f6118baaa0b11","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262058,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":8,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=8. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 8, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0059","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67a86116239edbd23aaca7c424e2cb599a4d92a0a428b08f9674f07b792f0126","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262059,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":9,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=9. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 9, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0060","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7f3769b470b412e2057f9c3309587405377e74cf0f1310b2b687f5a4667ec9","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262060,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":10,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=10. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 10, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0061","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71a491650d99b2bb0f2a7b58300edc49be29e5b75dae3e451165d01186851173","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262061,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":11,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=11. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 11, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 11, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0062","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739aca7668922fd1aa5f206061d50981e92738925ef3c6680d3957c3b38a08b8","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262062,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":12,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=12. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 12, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0063","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5abccb2ab03f7bc04489e04a17b99753adc857ac70138095673f70f65f7a6051","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262063,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":13,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=13. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 13, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 13, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0064","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbbbfec72c8c01f6ec9b5a14483ea1bfe27fdf892f76dfde3c07ceb144aabb5b","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262064,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":14,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=14. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 14, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0065","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45bbda713db384384662733162034d7ed90fadd8c27ded53bae0a2ae473af8a6","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262065,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":15,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=15. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 15, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0066","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10b8486f7ac513651c42df1d0d21e3546425f75bb950d24c80aa7fc9a6f8cd91","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262066,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":16,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=16. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 16, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 16, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0067","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4995b25281baeadbfdb047198a04fa22af045f270aad5444a2bcc144fa3425fa","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262067,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":17,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=17. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 17, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0068","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7516c429fc57103a862474b0753b760efbc3b5e83c138a0bfe525647f30a123","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262068,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":18,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=18. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 18, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0069","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a99be97d3c17482286ea21584b1e886281afdb28b2334b44866d404c682547b9","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262069,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":19,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=19. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 19, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0070","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70be8b6be3ae77474eee00515697d8eb179ccf6e518af3f8da48b5718f86dc1e","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262070,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":20,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=20. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 20, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 20, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0071","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdd59a428fab845398e76653834977b802f2cbe1897454eb68dc3e9074b361e7","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262071,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":21,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=21. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 21, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 21, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0072","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"297592d57bff6ada3e34c8ef362dbb97b11bcf0ea44252b2c2aaadbbad47a61a","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262072,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":22,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=22. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 22, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0073","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecaf2106e2918e1341003261da9fe723412c3cd80d70522bf61220be8950fd86","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262073,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":23,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=23. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 23, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 23, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0074","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5ce08916633ff3718b09ab1a0fc8bc60e7c25ddc735dc26b0f2151df67f0c5","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262074,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":24,\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=24. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(linfinity, 24, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 24, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0075","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66cac14e85950c4b75684d44a09512630bf19cb5b32ecd2d3886ab1f4fabcebe","family_id":"qf-mod26-norm-named-norm-certificate","seed":202610262075,"semantic_template":"qf-mod26-norm-named-norm-certificate/{\"dimension\":25,\"norm_name\":\"l1\"}"},"fields":{"title":"A Named-Formula Certificate Is Universal, Not Sampled","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=25. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).","answer":{"kind":"expression","value":"(l1, 25, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)","explanation":"The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 25, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)."},"skills":["A Named-Formula Certificate Is Universal, Not Sampled"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0076","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5434b9c3a2409610e93438ed43f56ec9cab8571b86dd4a532db156c1e891fd6","family_id":"qf-mod26-norm-induced-distance","seed":202610262076,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-12,-3,-5],\"norm_name\":\"l1\",\"right\":[12,-6,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0077","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e5e91bd1236fed06fb3ecd5bad4dcd2d7e1ee3ad9b5bbe12ab742a0351a315","family_id":"qf-mod26-norm-induced-distance","seed":202610262077,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-11,-2,-2],\"norm_name\":\"linfinity\",\"right\":[11,-1,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0078","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2414c55aa78f3cb51a45556a3d07130ea2e18ed6e6b7d62142420c9af88ac516","family_id":"qf-mod26-norm-induced-distance","seed":202610262078,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-10,-1,1],\"norm_name\":\"l1\",\"right\":[10,4,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0079","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fbf34be16e5bca5bff733bc2f2bbf8ef6617d12f163ca6be7cfd3818943a1e5","family_id":"qf-mod26-norm-induced-distance","seed":202610262079,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-9,0,4],\"norm_name\":\"linfinity\",\"right\":[9,-4,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0080","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be4a5893e8d21292c90c841c38b760dd2bb8e97d660144f8cc6580990647d8b","family_id":"qf-mod26-norm-induced-distance","seed":202610262080,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-8,1,-4],\"norm_name\":\"l1\",\"right\":[8,1,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0081","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bbf8aba01495c0134569cfe30ea67994d36d0851b3926fdf19549a4d6527754","family_id":"qf-mod26-norm-induced-distance","seed":202610262081,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-7,2,-1],\"norm_name\":\"linfinity\",\"right\":[7,6,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0082","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e561e9dda64401278c43bde1ceba3acbfbcc56526b506643c7d5a10bf50817fc","family_id":"qf-mod26-norm-induced-distance","seed":202610262082,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-6,3,2],\"norm_name\":\"l1\",\"right\":[6,-2,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0083","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5674c40526640e5ffc6e70ad6c01fadff8e3bda1fa6b19e65e16e864cd41faf2","family_id":"qf-mod26-norm-induced-distance","seed":202610262083,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-5,-3,5],\"norm_name\":\"linfinity\",\"right\":[5,3,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0084","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b9b7203b3ee3edc67190474c6372e5124a3e9f8d8ca10c3434869490f378ebd","family_id":"qf-mod26-norm-induced-distance","seed":202610262084,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-4,-2,-3],\"norm_name\":\"l1\",\"right\":[4,-5,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0085","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbbd754f440426b3ca879d74dba824d49a4eda05cdf541eef320dd2dcbae8e03","family_id":"qf-mod26-norm-induced-distance","seed":202610262085,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-3,-1,0],\"norm_name\":\"linfinity\",\"right\":[3,0,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0086","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91c635f84a36e86ce83e1c83c6fb6014db19fa51f0ad18e13236235f0adcaad2","family_id":"qf-mod26-norm-induced-distance","seed":202610262086,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-2,0,3],\"norm_name\":\"l1\",\"right\":[2,5,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0087","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9912f53f483997cabb487fd0199a46ef18ca4b08de82f4d65c5b30bad521deb8","family_id":"qf-mod26-norm-induced-distance","seed":202610262087,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[-1,1,-5],\"norm_name\":\"linfinity\",\"right\":[1,-3,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0088","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b20f11e9b28e85285cd35e6659a38055378c23486ce9966befb16172c0ce46f","family_id":"qf-mod26-norm-induced-distance","seed":202610262088,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[0,2,-2],\"norm_name\":\"l1\",\"right\":[0,2,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0089","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b837a20b9b196ef9dd6b8e9f5f2f189232b0c23c99ef2f2037f9dd9abcaf438d","family_id":"qf-mod26-norm-induced-distance","seed":202610262089,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[1,3,1],\"norm_name\":\"linfinity\",\"right\":[-1,-6,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0090","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edadd70a92e96d4cad8c244b0d05ce43926c41303a39fdc9c7d885e475b91ffb","family_id":"qf-mod26-norm-induced-distance","seed":202610262090,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[2,-3,4],\"norm_name\":\"l1\",\"right\":[-2,-1,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0091","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74a129ad058dff84f0bc60b7f4fe6af8f934f55a5397008b64fadc3c9b786911","family_id":"qf-mod26-norm-induced-distance","seed":202610262091,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[3,-2,-4],\"norm_name\":\"linfinity\",\"right\":[-3,4,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0092","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0245097a05dd23e4f9f81bcad149b95267ff0f13dd420cd028786935ce722581","family_id":"qf-mod26-norm-induced-distance","seed":202610262092,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[4,-1,-1],\"norm_name\":\"l1\",\"right\":[-4,-4,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0093","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40960d8b5bb350e6f5bac09a732ecbd1cb876a5b3f7d8bc014b87461a4541bad","family_id":"qf-mod26-norm-induced-distance","seed":202610262093,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0094","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c091521d57c25317dc95d1281dad64b62ddf278f753570361c7c800c1f1514f","family_id":"qf-mod26-norm-induced-distance","seed":202610262094,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[6,1,5],\"norm_name\":\"l1\",\"right\":[-6,6,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0095","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50a72078f8e65d7826bb71b79d15e487d0984c5efc755790f9d017f017af68c0","family_id":"qf-mod26-norm-induced-distance","seed":202610262095,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[7,2,-3],\"norm_name\":\"linfinity\",\"right\":[-7,-2,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0096","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feba4c3d4edcb663df6704e5f296b542cd86f3487cee7e1cb417c4135f13881a","family_id":"qf-mod26-norm-induced-distance","seed":202610262096,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[8,3,0],\"norm_name\":\"l1\",\"right\":[-8,3,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0097","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7caaf58aedf5a62e08462bd74300fe38b151246dddb5021fe6f1fb938c40fa8","family_id":"qf-mod26-norm-induced-distance","seed":202610262097,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[9,-3,3],\"norm_name\":\"linfinity\",\"right\":[-9,-5,-1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0098","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ebcf3f83957ecc752694445117b883e0b93ebfcc75b4275195e80a13e193ce","family_id":"qf-mod26-norm-induced-distance","seed":202610262098,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[10,-2,-5],\"norm_name\":\"l1\",\"right\":[-10,0,0]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0099","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8fe40e31a694270ccb2307611530a10181bd6c53362dc8c291fb8ae2b1e286b","family_id":"qf-mod26-norm-induced-distance","seed":202610262099,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r02-real-normed-vector-space-0100","batch_id":"batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dceb269d14fe1fcedad44251b7576e9085d251b2487bfbe5f9519496ea21a1c","family_id":"qf-mod26-norm-induced-distance","seed":202610262100,"semantic_template":"qf-mod26-norm-induced-distance/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"},"fields":{"title":"The Norm of an Ordered Vector Difference","stem":"Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).","answer":{"kind":"expression","value":"(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30)","explanation":"Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30)."},"skills":["The Norm of an Ordered Vector Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}