{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0001","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a55da1091419da1f81cb9520f4e3c1bf0f008d8d7380699632e4f68e06d7467","family_id":"qf-mod27-normmetric-distance","seed":202610272001,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-12,-3,-5],\"norm_name\":\"l1\",\"right\":[12,-6,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0002","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce0f51b76f9c37c05e3f1af4ba5e7db9440227982ec8f20450e06f9799bd59e","family_id":"qf-mod27-normmetric-distance","seed":202610272002,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-11,-2,-2],\"norm_name\":\"linfinity\",\"right\":[11,-1,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0003","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41976e89e10520a131a058228c8c03f937fc02db13a8ce622c331f63cb21991c","family_id":"qf-mod27-normmetric-distance","seed":202610272003,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-10,-1,1],\"norm_name\":\"l1\",\"right\":[10,4,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0004","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b53953fc4704e266bc999fc37bcc01d6a7f4db6c3a9614d7a97d02e2aa1bf2bf","family_id":"qf-mod27-normmetric-distance","seed":202610272004,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-9,0,4],\"norm_name\":\"linfinity\",\"right\":[9,-4,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0005","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b48ba9af7415816c96f6448f78bec9bad36e22cacebd60843d50d25afcb75f7","family_id":"qf-mod27-normmetric-distance","seed":202610272005,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-8,1,-4],\"norm_name\":\"l1\",\"right\":[8,1,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0006","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3274356a4e4add81e4eb27d51c013906b13ecdffdbe6f708890c0a4d8225ea4","family_id":"qf-mod27-normmetric-distance","seed":202610272006,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-7,2,-1],\"norm_name\":\"linfinity\",\"right\":[7,6,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0007","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"848f863234d1f8adc7d8eeac7f8037269a346adb18ad346bec45a0fb40cd9289","family_id":"qf-mod27-normmetric-distance","seed":202610272007,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-6,3,2],\"norm_name\":\"l1\",\"right\":[6,-2,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0008","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90ffd7aa2aef99634e8e40124c8e717865524274b716620191b38605130c4cd8","family_id":"qf-mod27-normmetric-distance","seed":202610272008,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-5,-3,5],\"norm_name\":\"linfinity\",\"right\":[5,3,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0009","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c7d6d080168e9aed651b4546a52a8e5c4331538cf5b0200977c2e711cc51eb9","family_id":"qf-mod27-normmetric-distance","seed":202610272009,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-4,-2,-3],\"norm_name\":\"l1\",\"right\":[4,-5,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0010","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44c0f8fbf20338a539935b1f9ad9a84f3cdc0dda6aa16dc1d447b3923f0ee064","family_id":"qf-mod27-normmetric-distance","seed":202610272010,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-3,-1,0],\"norm_name\":\"linfinity\",\"right\":[3,0,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0011","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc94616e70527c10c23402350310a627dca53394000e639ad7f8eec6bd800d81","family_id":"qf-mod27-normmetric-distance","seed":202610272011,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-2,0,3],\"norm_name\":\"l1\",\"right\":[2,5,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0012","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b5b3de4be0f993cac26ea475b8bcfdf9868f7836419800e617b42f3ae520caf","family_id":"qf-mod27-normmetric-distance","seed":202610272012,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[-1,1,-5],\"norm_name\":\"linfinity\",\"right\":[1,-3,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0013","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa5c9da8355e940cfa01a324818f84e48198bc29a936eedfd46abe1c4da1cf41","family_id":"qf-mod27-normmetric-distance","seed":202610272013,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[0,2,-2],\"norm_name\":\"l1\",\"right\":[0,2,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0014","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6688c4b9bdd7e71b3580553e25fc4de8db67246078fc5d9aeba8bc737ec93d14","family_id":"qf-mod27-normmetric-distance","seed":202610272014,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[1,3,1],\"norm_name\":\"linfinity\",\"right\":[-1,-6,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0015","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95314769231ad96aa6f59d16b16b367795c7b476f4335684fc4c8de6b5a0ba3b","family_id":"qf-mod27-normmetric-distance","seed":202610272015,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[2,-3,4],\"norm_name\":\"l1\",\"right\":[-2,-1,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0016","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb6901f155ba8a54e2cfd0ebaa27e7ea570d4e2753731d1a75e0054fa557832","family_id":"qf-mod27-normmetric-distance","seed":202610272016,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[3,-2,-4],\"norm_name\":\"linfinity\",\"right\":[-3,4,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0017","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbc5a7deae91412108c2d3fd2371122c4adb77f1f841a33220444aa545d8c06f","family_id":"qf-mod27-normmetric-distance","seed":202610272017,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[4,-1,-1],\"norm_name\":\"l1\",\"right\":[-4,-4,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0018","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad7a1bb4abaf03765810be9495686559e5275cb2973a8d54ba7d4ab2f667887e","family_id":"qf-mod27-normmetric-distance","seed":202610272018,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0019","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ffd9c21572383021c2ce5b7226d09cf682b7154dfa8d7cdf6ccb4ae4ec70620","family_id":"qf-mod27-normmetric-distance","seed":202610272019,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[6,1,5],\"norm_name\":\"l1\",\"right\":[-6,6,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0020","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bca77962bf068d77540a86379606c85f9238649fc47c914f9a097903b6146b92","family_id":"qf-mod27-normmetric-distance","seed":202610272020,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[7,2,-3],\"norm_name\":\"linfinity\",\"right\":[-7,-2,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0021","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219540392fde6c61a0314117e153ac26c75558772e780a221e191521ad324228","family_id":"qf-mod27-normmetric-distance","seed":202610272021,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[8,3,0],\"norm_name\":\"l1\",\"right\":[-8,3,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0022","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e61018f19fe0be8117c4189dcb927aa1c0b42c1b68f5efdce215a34443e32138","family_id":"qf-mod27-normmetric-distance","seed":202610272022,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[9,-3,3],\"norm_name\":\"linfinity\",\"right\":[-9,-5,-1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0023","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61bb49c824a5e4076ad0ef27af0811b31671cdd78c9744851458f4f211ac4099","family_id":"qf-mod27-normmetric-distance","seed":202610272023,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[10,-2,-5],\"norm_name\":\"l1\",\"right\":[-10,0,0]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0024","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf81737cded244f1ae99dc3dd26f46d3029777091e9fbdb9fc270b353682f9af","family_id":"qf-mod27-normmetric-distance","seed":202610272024,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0025","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99e8dfd80140099124951623c940bdc772554a20621e55603fd7c09b925ff38f","family_id":"qf-mod27-normmetric-distance","seed":202610272025,"semantic_template":"qf-mod27-normmetric-distance/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"},"fields":{"title":"The Metric Induced by a Fixed Named Norm","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2]. 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 in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate 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 Metric Induced by a Fixed Named Norm"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0026","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"706c0327c9170e386d308999752cdc3bab158cc0877c4d094217060ac45d91d0","family_id":"qf-mod27-normmetric-symmetry","seed":202610272026,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-12,-3,-5],\"norm_name\":\"l1\",\"right\":[12,-6,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 30, 30, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 30, 30, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0027","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4247056f8607340bc5a8466a93e42e44daf9d5b61685a34748ece05b0f41a15a","family_id":"qf-mod27-normmetric-symmetry","seed":202610272027,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-11,-2,-2],\"norm_name\":\"linfinity\",\"right\":[11,-1,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 22, 22, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, 22, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0028","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab81bc5be790d665cc51f94a2643c625a7d5a88e8df9dd65d7fe07362121c36","family_id":"qf-mod27-normmetric-symmetry","seed":202610272028,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-10,-1,1],\"norm_name\":\"l1\",\"right\":[10,4,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 26, 26, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 26, 26, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0029","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a56ee10a55b72cb4c29e626d1e89d7cf1869ea2feaf95fa9c83dba43ec986de2","family_id":"qf-mod27-normmetric-symmetry","seed":202610272029,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-9,0,4],\"norm_name\":\"linfinity\",\"right\":[9,-4,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 18, 18, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, 18, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0030","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9208ecbda77269f78b0982a2bee2174bff80dbe0e3bc2007737102ee37c14322","family_id":"qf-mod27-normmetric-symmetry","seed":202610272030,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-8,1,-4],\"norm_name\":\"l1\",\"right\":[8,1,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 18, 18, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 18, 18, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0031","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04f5b7e5179e4071ab42ec7c9d426ea3289978d7b6c3b68ece724b113936542f","family_id":"qf-mod27-normmetric-symmetry","seed":202610272031,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-7,2,-1],\"norm_name\":\"linfinity\",\"right\":[7,6,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 14, 14, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0032","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8ebf1048cd35b8cfe6699c78bca8844aba583211d8dea251c2dce5c4c7b9519","family_id":"qf-mod27-normmetric-symmetry","seed":202610272032,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-6,3,2],\"norm_name\":\"l1\",\"right\":[6,-2,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 19, 19, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, 19, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0033","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f5bca9a0632867b129d1168eb79e806de042258863b64c954757b7513ca6cc","family_id":"qf-mod27-normmetric-symmetry","seed":202610272033,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-5,-3,5],\"norm_name\":\"linfinity\",\"right\":[5,3,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 10, 10, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 10, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0034","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29b3fc847fd4e79745da9db05064b0597217624f4b94f288700b62c940411947","family_id":"qf-mod27-normmetric-symmetry","seed":202610272034,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-4,-2,-3],\"norm_name\":\"l1\",\"right\":[4,-5,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 12, 12, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0035","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"867df2f94bf4d0b5e1c767ec3de04d31738eafcc140cd988716da280b523dd8f","family_id":"qf-mod27-normmetric-symmetry","seed":202610272035,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-3,-1,0],\"norm_name\":\"linfinity\",\"right\":[3,0,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 6, 6, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0036","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5d56892b3cf129eafc470f8371f1156a0b34d81359776c53ec92ec251c12468","family_id":"qf-mod27-normmetric-symmetry","seed":202610272036,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-2,0,3],\"norm_name\":\"l1\",\"right\":[2,5,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 12, 12, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0037","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ff75dc58f6c5df0cf0cdc2c72b8bcf83b2b42101162f1b6dd7b499ddc028a1b","family_id":"qf-mod27-normmetric-symmetry","seed":202610272037,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[-1,1,-5],\"norm_name\":\"linfinity\",\"right\":[1,-3,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 6, 6, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0038","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee164f2e1e06b76bbf4167922cd75350a3473dd55d71ac63b1b760696dc9e18e","family_id":"qf-mod27-normmetric-symmetry","seed":202610272038,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[0,2,-2],\"norm_name\":\"l1\",\"right\":[0,2,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 0, 0, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 0, 0, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0039","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ceba3ab152e27cf815c856ed23463b7bc2ed5569a4d340e7a67e47a25a860af","family_id":"qf-mod27-normmetric-symmetry","seed":202610272039,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[1,3,1],\"norm_name\":\"linfinity\",\"right\":[-1,-6,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 9, 9, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 9, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0040","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79be461efd4d02dac8420927d33b2413706cb2da85709970c42d4b38ecde5df4","family_id":"qf-mod27-normmetric-symmetry","seed":202610272040,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[2,-3,4],\"norm_name\":\"l1\",\"right\":[-2,-1,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 10, 10, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 10, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0041","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da056009645b5ecbe224c67b15a962a3d3ec2cbfc6ecaa47e06ce71ffe5c2756","family_id":"qf-mod27-normmetric-symmetry","seed":202610272041,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[3,-2,-4],\"norm_name\":\"linfinity\",\"right\":[-3,4,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 6, 6, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0042","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0466ce517ed4e83f760f06b0b5b677bc9ab7feabe648776851ff461d98180627","family_id":"qf-mod27-normmetric-symmetry","seed":202610272042,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[4,-1,-1],\"norm_name\":\"l1\",\"right\":[-4,-4,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 12, 12, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0043","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"079f19cd3f5dec18e983f4d459a37fe238d0e6458c5ea6ea0381486c23a8abda","family_id":"qf-mod27-normmetric-symmetry","seed":202610272043,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 10, 10, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 10, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0044","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b735f9727e00f96cb440d66a8550fba2290bb9752fb6d6e8c630724a4e78c50","family_id":"qf-mod27-normmetric-symmetry","seed":202610272044,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[6,1,5],\"norm_name\":\"l1\",\"right\":[-6,6,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 22, 22, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 22, 22, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0045","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f67923c4aeb941728a56ef56b0c5ff5dd636d86e650be0b8edda5e0d2052d3a5","family_id":"qf-mod27-normmetric-symmetry","seed":202610272045,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[7,2,-3],\"norm_name\":\"linfinity\",\"right\":[-7,-2,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 14, 14, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0046","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3161276dc93c17854e61f1fac28b65b94031a2dded2de680a55e636a44ef1c5","family_id":"qf-mod27-normmetric-symmetry","seed":202610272046,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[8,3,0],\"norm_name\":\"l1\",\"right\":[-8,3,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 18, 18, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 18, 18, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0047","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51ebbe6a9a4ca021f480155a4af4f7424fa8dbb48a6c2726f9b4c790267a8d00","family_id":"qf-mod27-normmetric-symmetry","seed":202610272047,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[9,-3,3],\"norm_name\":\"linfinity\",\"right\":[-9,-5,-1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 18, 18, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, 18, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0048","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45e01ddcfd671e20a97c039d78dd4b80911dbc0cd83a7b06c9e0782f722b1bc3","family_id":"qf-mod27-normmetric-symmetry","seed":202610272048,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[10,-2,-5],\"norm_name\":\"l1\",\"right\":[-10,0,0]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 27, 27, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 27, 27, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0049","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e7ba3b09d0829b505d0cd550ecd93c7c92dfc3bd1a3817638ba33e0eb20bfa1","family_id":"qf-mod27-normmetric-symmetry","seed":202610272049,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(linfinity, 22, 22, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, 22, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0050","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c77760f0c57b6a511a5dca0696537d6633da9ceea46f6ff143d40ed7d1272cfd","family_id":"qf-mod27-normmetric-symmetry","seed":202610272050,"semantic_template":"qf-mod27-normmetric-symmetry/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"},"fields":{"title":"Reverse the Difference, Not the Norm Definition","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,d_xy,d_yx,equal?).","answer":{"kind":"expression","value":"(l1, 30, 30, Yes)","explanation":"Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 30, 30, Yes)."},"skills":["Reverse the Difference, Not the Norm Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0051","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caf32d21dcbecb01d14df411e30d105cb51db76e81a4707888ebd80a1272faac","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272051,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-12,-3],\"y\":[-4,12],\"z\":[-6,-2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-12, -3], y=[-4, 12], z=[-6, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 7, 23, 16, 39, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, 23, 16, 39, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0052","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e04ad5bd079f0d3655f7f04adbe0e1bd3166c7686ca19c0d8c333db498b1ae5","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272052,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-11,-2],\"y\":[-3,11],\"z\":[-1,-1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-11, -2], y=[-3, 11], z=[-1, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 10, 13, 12, 25, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 13, 12, 25, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0053","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb0718a465ea48079941d09fae9dfdca86793a7e43b770f3d438b5e3472b6f51","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272053,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-10,-1],\"y\":[-2,10],\"z\":[4,0]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-10, -1], y=[-2, 10], z=[4, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 15, 19, 16, 35, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, 19, 16, 35, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0054","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c312d54b92ed47c4f647bc6bc83d5100899573934db6a6378c0ff2faf23f187f","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272054,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-9,0],\"y\":[-1,9],\"z\":[-4,1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-9, 0], y=[-1, 9], z=[-4, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 5, 9, 8, 17, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 5, 9, 8, 17, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0055","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05fd5ec69ded68aa8aef2e74d415c326bd4e4a4e0076ce78bcb7adb9e2f28e3e","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272055,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-8,1],\"y\":[0,8],\"z\":[1,2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-8, 1], y=[0, 8], z=[1, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 10, 15, 7, 22, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 15, 7, 22, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0056","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"949d4e98888447bf95f94593fe3c800279197d12d2d50f8394e9da523884236a","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272056,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-7,2],\"y\":[1,7],\"z\":[6,-2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-7, 2], y=[1, 7], z=[6, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 13, 8, 9, 17, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 13, 8, 9, 17, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0057","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"000a26dbf8e7602da725f52d746762dda9d236123457e5ae4e5b824c11506c23","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272057,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-6,3],\"y\":[2,6],\"z\":[-2,-1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-6, 3], y=[2, 6], z=[-2, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 8, 11, 11, 22, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 11, 11, 22, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0058","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7bbb59a290802ceda2313b4d1f769a5a41ba91359cba27555179d50dd6150ac","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272058,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-5,-3],\"y\":[3,5],\"z\":[3,0]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-5, -3], y=[3, 5], z=[3, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 8, 8, 5, 13, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, 8, 5, 13, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0059","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d72b49c1f9c85e0c7d991beadfba78417ad485331ad0afb59373f7a05009de2b","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272059,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-4,-2],\"y\":[4,4],\"z\":[-5,1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-4, -2], y=[4, 4], z=[-5, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 4, 14, 12, 26, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 4, 14, 12, 26, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0060","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18816146e7eec7dd9be337504bed4c14a074648004e20c30f9ca72ee1f914728","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272060,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-3,-1],\"y\":[-4,3],\"z\":[0,2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-3, -1], y=[-4, 3], z=[0, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 3, 4, 4, 8, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 3, 4, 4, 8, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0061","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f50dab9555714d5ce7731f9e11da69cda68d48b4045d9d27ae69b1af64973f2d","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272061,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[-2,0],\"y\":[-3,2],\"z\":[5,-2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-2, 0], y=[-3, 2], z=[5, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 9, 3, 12, 15, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, 3, 12, 15, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0062","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea4df6c14855c3fdd37636693f32f199cfccdd9631eb4bdf31155a652c546270","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272062,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[-1,1],\"y\":[-2,1],\"z\":[-3,-1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-1, 1], y=[-2, 1], z=[-3, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 2, 1, 2, 3, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, 1, 2, 3, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0063","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d4ce03d2a94e787469154c7dee9f9e49e1eadf4d99bf21b929a24b6d3d28ac0","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272063,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[0,2],\"y\":[-1,0],\"z\":[2,0]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[0, 2], y=[-1, 0], z=[2, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 4, 3, 3, 6, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 4, 3, 3, 6, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0064","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5451b8c5ea62e4a7c0d56576a8b3d3b57c8194374833e1a507fdb6a14b703ab4","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272064,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[1,3],\"y\":[0,-1],\"z\":[-6,1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[1, 3], y=[0, -1], z=[-6, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 7, 4, 6, 10, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 4, 6, 10, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0065","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ffeb219c44d7035ff33c1ede8a4599c7dc042276b1bd63ca9cc7bee6c985a0","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272065,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[2,-3],\"y\":[1,-2],\"z\":[-1,2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[2, -3], y=[1, -2], z=[-1, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 8, 2, 6, 8, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 2, 6, 8, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0066","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d3c277948612950a0f637100765f61ee85018d3f979a1efbca76b934bc806d","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272066,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[3,-2],\"y\":[2,-3],\"z\":[4,-2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[3, -2], y=[2, -3], z=[4, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 1, 1, 2, 3, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 1, 1, 2, 3, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0067","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ab2cac81b1f5c374dc96cb6c4b89e20f8f9d933de81ee2aa9a97ff23713ad69","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272067,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[4,-1],\"y\":[3,-4],\"z\":[-4,-1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[4, -1], y=[3, -4], z=[-4, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 8, 4, 10, 14, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 4, 10, 14, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0068","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06023c88329ccb0e865d3798b8e7af327b22fb5962de068d635ab136a7ea4ffd","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272068,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[5,0],\"y\":[4,-5],\"z\":[1,0]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[5, 0], y=[4, -5], z=[1, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 4, 5, 5, 10, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, 5, 5, 10, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0069","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f05786c2496434cdfd82187be97567f3fa2dabdf196f13cb52088933ede6222","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272069,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[6,1],\"y\":[-4,-6],\"z\":[6,1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[6, 1], y=[-4, -6], z=[6, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 0, 17, 17, 34, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 0, 17, 17, 34, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0070","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"835f9febd3074b64933f6bcc5cf994466182fc286538ec21b21e7188df1f8ef8","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272070,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[7,2],\"y\":[-3,-7],\"z\":[-2,2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[7, 2], y=[-3, -7], z=[-2, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 9, 10, 9, 19, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 10, 9, 19, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0071","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b30fae2c5a9a5a1c8707375eda8c25e595920e540efbc30d6c4a670df242c471","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272071,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[8,3],\"y\":[-2,-8],\"z\":[3,-2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[8, 3], y=[-2, -8], z=[3, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 10, 21, 11, 32, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 21, 11, 32, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0072","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7235fbe9e1f74264645cc0ac261663428f33918c48a2475b5b7390c52b62956c","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272072,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[9,-3],\"y\":[-1,-9],\"z\":[-5,-1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[9, -3], y=[-1, -9], z=[-5, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 14, 10, 8, 18, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 10, 8, 18, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0073","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49cd5cc037d574bc3757b226b107f81257200fecb707529a35eb2b0c79a731df","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272073,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[10,-2],\"y\":[0,-10],\"z\":[0,0]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[10, -2], y=[0, -10], z=[0, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 12, 18, 10, 28, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 18, 10, 28, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0074","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66202312b4c62ef77794086641d23c9e977bb36f4ffe903c6abecd9e994ba232","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272074,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"linfinity\",\"x\":[11,-1],\"y\":[1,-11],\"z\":[5,1]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[11, -1], y=[1, -11], z=[5, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(linfinity, 6, 10, 12, 22, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 10, 12, 22, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0075","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2487ae60617faf42701a2f82d2c711cf94a6753a40a785acf94eb4e6fc21d6e4","family_id":"qf-mod27-normmetric-triangle-certificate","seed":202610272075,"semantic_template":"qf-mod27-normmetric-triangle-certificate/{\"norm_name\":\"l1\",\"x\":[12,0],\"y\":[2,-12],\"z\":[-3,2]}"},"fields":{"title":"The Norm Triangle Inequality Supplies the Metric Triangle Bound","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[12, 0], y=[2, -12], z=[-3, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).","answer":{"kind":"expression","value":"(l1, 17, 22, 19, 41, Yes, norm_triangle_inequality)","explanation":"Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, 22, 19, 41, Yes, norm_triangle_inequality)."},"skills":["The Norm Triangle Inequality Supplies the Metric Triangle Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0076","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf21a3bc569bb97ad188f52662f8d8f47717bd5f11b5db0decb8544384d581af","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272076,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":1,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=1, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 1, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 1, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0077","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13ccab2a86cfe8c1485d1da4b013f4cb3432a9dea58b6584a8a7af63e44ce1d5","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272077,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":2,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=2, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 2, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0078","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c99ab85ac063072e682d2c4c2f0bc9d0a1252cb4af94bf5b54492e7584c39a44","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272078,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":3,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=3, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 3, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 3, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0079","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dc687d0124dc59486dd12832c7c40471bd3d3e6374d2ff9beac8ee130cb7ef6","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272079,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":4,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=4, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 4, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0080","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c904b73bc463dd516b2e55c0aa7c905f7ea10776f18182bcdc1707922ec2925","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272080,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":5,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=5, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 5, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 5, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0081","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c83279002b3286b794f8405c98c3ab1cc5c42f5af8f63e29800b87a1c03ba3b","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272081,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":6,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=6, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 6, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0082","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aed2a967f29fb1c093da8a669fa5a3dc03d6d6ae641fec1a57d60d29b242abe8","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272082,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":7,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=7, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 7, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0083","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66c23ba8664311bf0030ed45959a06da919736d1578f9426441f60bb7ea5eb37","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272083,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":8,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=8, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 8, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0084","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d174b8c2b66151b4c2dbda606d5f8f7c7cf8eeffe05fca058eeea09214bf9447","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272084,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":9,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=9, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 9, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0085","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6c5a513cef4560bb994306e7d0eb681f3a4b876eab08e79ef35fb5cc039b107","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272085,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":10,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=10, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 10, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0086","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95b55b3628682277eb7b0318beb68db4219cbd919d7d2ea56caf220f2d10b0f","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272086,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":11,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=11, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 11, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 11, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0087","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d71143945de731afbb17d460d579c020c8e6c016f681528fd879d7e6835d6da6","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272087,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":12,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=12, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 12, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0088","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95fba641c547e055989dfcca50fd31ed862905ccf88b3907606b2ab6dbff35b2","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272088,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":13,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=13, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 13, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 13, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0089","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4f8c4c3d5e53751a7b308179867852aa301767f4fcb7dde1d1a3df6e3f0b3b","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272089,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":14,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=14, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 14, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0090","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b0bd89cd6a6ac6861dbcdb26d85837ad6345864a23a7824719051c7ff5151d","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272090,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":15,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=15, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0091","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"107c1ce53cd90d54d54ec8710186ea6bcc8abd1ab7f1fdf32e996d4401b339cd","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272091,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":16,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=16, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 16, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 16, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0092","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5086d86a71ab5d6fb727f83601c9acb7e691820076a86316daa8b6575f0fc900","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272092,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":17,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=17, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 17, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0093","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8250356029999c5166e7d8e2b676258bd5426561b3d4788ed48d2e2ea1e626ae","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272093,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":18,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=18, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 18, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0094","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0a70b5dfb81b839e5ce44bdc48ac0b2a0ea6737e1a23ce24df7c7066978fb94","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272094,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":19,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=19, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 19, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0095","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95dd3d0b79cf444eef1467913b9c8b95f4ff0b4e2cb6803430d1301e7a96b5ea","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272095,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":20,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=20, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 20, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 20, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0096","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faa60700ff7f44aebec48b1db024703c0df0a2e86a36c3a537fa4432a3dbfdf9","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272096,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":21,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=21, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 21, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 21, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0097","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9cdd7a6e4ff046b2d4d093199cc6765d2072b0c1b6ef5a1cf57a7029b978646","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272097,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":22,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=22, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 22, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0098","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"769ef986d957394afbaaaef78d546bdb473e43b9580aa8a511cf417daedc4b58","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272098,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":23,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=23, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 23, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 23, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0099","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddfa686824d4b13a4a57f2f6827ef9b4e27c6ad4cf503d5f9d75040c548f090e","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272099,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":24,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"linfinity\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=24, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(linfinity, 24, named_formula_certificate, Yes, universal_from_norm_axioms)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 24, named_formula_certificate, Yes, universal_from_norm_axioms)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r02-norm-induced-metric-0100","batch_id":"batch-20261014-advance-20261027-r02-norm-induced-metric-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eca30b89f4e41c9c4fdd2804911bce753839a65675382981d254ca7ae729cb9","family_id":"qf-mod27-normmetric-induced-qualification","seed":202610272100,"semantic_template":"qf-mod27-normmetric-induced-qualification/{\"dimension\":25,\"evidence_mode\":\"finite_samples\",\"norm_name\":\"l1\"}"},"fields":{"title":"Distinguish a Universal Norm Certificate from Finite Samples","stem":"Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=25, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.","answer":{"kind":"expression","value":"(l1, 25, finite_samples, No, samples_do_not_establish_induced_metric)","explanation":"The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 25, finite_samples, No, samples_do_not_establish_induced_metric)."},"skills":["Distinguish a Universal Norm Certificate from Finite Samples"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}