{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0001","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c605f8ccf09d8680d13745ead52ea04b6c3d5383ed3d5ff376c20aa96eaa4e22","family_id":"qf-mod28-operator-l1-norm","seed":202610285001,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-12,-3,-5],[-2,12,-6]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-12, -3, -5], [-2, 12, -6]], [14, 15, 11], 15, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [14, 15, 11], 15, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0002","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c1decf9f27fc59d6ba05180794095517984fe2e870f7381564664d8bd1fc1ab","family_id":"qf-mod28-operator-l1-norm","seed":202610285002,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-11,-2,-2],[-1,11,-1]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-11, -2, -2], [-1, 11, -1]], [12, 13, 3], 13, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [12, 13, 3], 13, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0003","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfeb1d6639afeb40bdcbb7d5d9c83e84427adfa8a8758dfcf422aeba3ad96449","family_id":"qf-mod28-operator-l1-norm","seed":202610285003,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-10,-1,1],[0,10,4]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-10, -1, 1], [0, 10, 4]], [10, 11, 5], 11, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [10, 11, 5], 11, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0004","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c85887bb7935bf64743c74d0a71eff30f1de854e9a7d20f8280180b61326e22","family_id":"qf-mod28-operator-l1-norm","seed":202610285004,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-9,0,4],[1,9,-4]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-9, 0, 4], [1, 9, -4]], [10, 9, 8], 10, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [10, 9, 8], 10, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0005","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff4e94fa43c1d0ed5dfbeb6938d211161cb20b9ee6ded5470a2a12b75e2eedf8","family_id":"qf-mod28-operator-l1-norm","seed":202610285005,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-8,1,-4],[2,8,1]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-8, 1, -4], [2, 8, 1]], [10, 9, 5], 10, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [10, 9, 5], 10, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0006","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e05e521d8da0a33ec055e9492bf5a7e2055ebc553b0d7ef2c3ea164d1efe1d4","family_id":"qf-mod28-operator-l1-norm","seed":202610285006,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-7,2,-1],[-2,7,6]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-7, 2, -1], [-2, 7, 6]], [9, 9, 7], 9, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [9, 9, 7], 9, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0007","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e55bb7f52f0f92cc6e80116ab4a6ca8dcdb1eb439235af13f6d8446bca7fa9c0","family_id":"qf-mod28-operator-l1-norm","seed":202610285007,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-6,3,2],[-1,6,-2]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-6, 3, 2], [-1, 6, -2]], [7, 9, 4], 9, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [7, 9, 4], 9, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0008","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5355c44985d7a190bec31943f5802e4134e2b26d9660334f21104d0806f14bc","family_id":"qf-mod28-operator-l1-norm","seed":202610285008,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-5,-3,5],[0,5,3]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-5, -3, 5], [0, 5, 3]], [5, 8, 8], 8, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [5, 8, 8], 8, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0009","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e8bbd8de3c9f2f59bc7d2b114433e222c98f31e827e1c0d0ac6d87beaded1c9","family_id":"qf-mod28-operator-l1-norm","seed":202610285009,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-4,-2,-3],[1,4,-5]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-4, -2, -3], [1, 4, -5]], [5, 6, 8], 8, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [5, 6, 8], 8, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0010","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f5be12b825fabc5a438d4ef4dd75535e73d792d09ba35f220c366130ed1f688","family_id":"qf-mod28-operator-l1-norm","seed":202610285010,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-3,-1,0],[2,3,0]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-3, -1, 0], [2, 3, 0]], [5, 4, 0], 5, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [5, 4, 0], 5, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0011","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c088dfd85a4b6222fc1a3e3a036552f03c0343fe65fc49ce34e333f9f728cfad","family_id":"qf-mod28-operator-l1-norm","seed":202610285011,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-2,0,3],[-2,2,5]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-2, 0, 3], [-2, 2, 5]], [4, 2, 8], 8, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [4, 2, 8], 8, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0012","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3cef969b0dd57cc5da1c0ac87d1c9ac9887bf981589649d03a9eddb93f990b1","family_id":"qf-mod28-operator-l1-norm","seed":202610285012,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[-1,1,-5],[-1,1,-3]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[-1, 1, -5], [-1, 1, -3]], [2, 2, 8], 8, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [2, 2, 8], 8, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0013","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72c979d9f23d6e08ddf1b1ce7e1ffabe81100bbc72d9c1753f8832a822e7ec71","family_id":"qf-mod28-operator-l1-norm","seed":202610285013,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[0,2,-2],[0,0,2]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[0, 2, -2], [0, 0, 2]], [0, 2, 4], 4, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [0, 2, 4], 4, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0014","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84cb02bc6e908f3cc4e03205fa680daee1f7914dad99664b9abb102ecc84577b","family_id":"qf-mod28-operator-l1-norm","seed":202610285014,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[1,3,1],[1,-1,-6]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[1, 3, 1], [1, -1, -6]], [2, 4, 7], 7, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [2, 4, 7], 7, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0015","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f3e9ef855e9ec335eea94e4f87ec2647a1964f0f3908f33d96ed7bd52f1b072","family_id":"qf-mod28-operator-l1-norm","seed":202610285015,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[2,-3,4],[2,-2,-1]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[2, -3, 4], [2, -2, -1]], [4, 5, 5], 5, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [4, 5, 5], 5, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0016","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9146ea73640a93edf5972d1ca0c77e9bb705b63a69e59e0e982928a323c2398d","family_id":"qf-mod28-operator-l1-norm","seed":202610285016,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[3,-2,-4],[-2,-3,4]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[3, -2, -4], [-2, -3, 4]], [5, 5, 8], 8, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [5, 5, 8], 8, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0017","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d416105a4a5ed5ee82ada10448588926f8859e05bb7d1a56d9f9dbdbe26986ee","family_id":"qf-mod28-operator-l1-norm","seed":202610285017,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[4,-1,-1],[-1,-4,-4]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[4, -1, -1], [-1, -4, -4]], [5, 5, 5], 5, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [5, 5, 5], 5, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0018","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efc7c83e36bf0fb0ab8b1a7037346b7b1ddfcb0512a64afd17e9e96fbadfeffc","family_id":"qf-mod28-operator-l1-norm","seed":202610285018,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[5,0,2],[0,-5,1]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[5, 0, 2], [0, -5, 1]], [5, 5, 3], 5, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [5, 5, 3], 5, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0019","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef08adbf95b30231f76857eafc5f569994c486010e477cf5cc0f8711b9ddb12d","family_id":"qf-mod28-operator-l1-norm","seed":202610285019,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[6,1,5],[1,-6,6]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[6, 1, 5], [1, -6, 6]], [7, 7, 11], 11, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [7, 7, 11], 11, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0020","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"572da725d33a5805f954a5a2238f55d47266ca021772ba13c20ee53f75bbb69c","family_id":"qf-mod28-operator-l1-norm","seed":202610285020,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[7,2,-3],[2,-7,-2]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[7, 2, -3], [2, -7, -2]], [9, 9, 5], 9, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [9, 9, 5], 9, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0021","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af66cb9d7ef843e7c346946e0537c0fe8f49902563c2fcccca1ca0a5489d8edd","family_id":"qf-mod28-operator-l1-norm","seed":202610285021,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[8,3,0],[-2,-8,3]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[8, 3, 0], [-2, -8, 3]], [10, 11, 3], 11, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [10, 11, 3], 11, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0022","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71adc41b11527e16a5938118b4d009e5688005cedd15d0d4b495a7a07fb1ccab","family_id":"qf-mod28-operator-l1-norm","seed":202610285022,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[9,-3,3],[-1,-9,-5]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[9, -3, 3], [-1, -9, -5]], [10, 12, 8], 12, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [10, 12, 8], 12, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0023","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fa993d99159cdb4a5bff6aabc0ef69639196cfefad8e2c1d14a6a04fb15961c","family_id":"qf-mod28-operator-l1-norm","seed":202610285023,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[10,-2,-5],[0,-10,0]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[10, -2, -5], [0, -10, 0]], [10, 12, 5], 12, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [10, 12, 5], 12, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0024","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce874f899fb8ca4b84c3a5c8fb069fded4ec1c179c6045d82020c1b56ff2b980","family_id":"qf-mod28-operator-l1-norm","seed":202610285024,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[11,-1,-2],[1,-11,5]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[11, -1, -2], [1, -11, 5]], [12, 12, 7], 12, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [12, 12, 7], 12, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0025","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30fa2607beec85ee2fab3cac5734593137cc2ee3a5d0ddbc277f2bd6e2d6d0b8","family_id":"qf-mod28-operator-l1-norm","seed":202610285025,"semantic_template":"qf-mod28-operator-l1-norm/{\"matrix\":[[12,0,1],[2,-12,-3]]}"},"fields":{"title":"The Exact l1 Operator Norm Uses Absolute Column Sums","stem":"Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.","answer":{"kind":"expression","value":"([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)","explanation":"Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)."},"skills":["The Exact l1 Operator Norm Uses Absolute Column Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0026","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd5c81de7d38ac335528d2fd646b5d016f31bf3eda058df460c66d1b1a043831","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285026,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-12,-3,-5],[-2,12,-6]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-12, -3, -5], [-2, 12, -6]], [20, 20], 20, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [20, 20], 20, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0027","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c1da180990248d4cc82c20d2424ef9ba6d1d72da39ee66982d7e721633705ce","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285027,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-11,-2,-2],[-1,11,-1]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-11, -2, -2], [-1, 11, -1]], [15, 13], 15, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [15, 13], 15, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0028","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"336a9f5419248ebbbe50359b9f9c8563bd6fe8ec592ec5cc0d362e84b3d3d4c2","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285028,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-10,-1,1],[0,10,4]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-10, -1, 1], [0, 10, 4]], [12, 14], 14, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [12, 14], 14, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0029","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74bd759077486335178778db860302ca0618fcb68fd455cb6b1b03770d0af39","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285029,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-9,0,4],[1,9,-4]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-9, 0, 4], [1, 9, -4]], [13, 14], 14, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [13, 14], 14, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0030","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bff0f45681e74ff879ab6093d64a283b7b0a17f277ac849c63549c653ec6c8d","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285030,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-8,1,-4],[2,8,1]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-8, 1, -4], [2, 8, 1]], [13, 11], 13, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [13, 11], 13, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0031","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fa4e2fcd447ccce2d7375f7b12ee8ec6a1b12cee450a5f280b7ebd381f64095","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285031,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-7,2,-1],[-2,7,6]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-7, 2, -1], [-2, 7, 6]], [10, 15], 15, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [10, 15], 15, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0032","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3606ac904324dd2ab32975080a97c365e1731cd2aa97e6de6ad9a7a47cd9c30a","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285032,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-6,3,2],[-1,6,-2]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-6, 3, 2], [-1, 6, -2]], [11, 9], 11, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [11, 9], 11, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0033","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c6b1998cc6d29ee982f91c9bd63669a1be6168ab1558eb08aba7f2d8f871dc6","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285033,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-5,-3,5],[0,5,3]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-5, -3, 5], [0, 5, 3]], [13, 8], 13, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [13, 8], 13, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0034","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"496ca849567b9b4b58d81568a48c9d83f51a2ef15c1511265d80d42948c5870f","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285034,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-4,-2,-3],[1,4,-5]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-4, -2, -3], [1, 4, -5]], [9, 10], 10, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [9, 10], 10, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0035","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e18af339496762417cc908e9dcec00646952e91475ae955dd59d89cc2c951ee5","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285035,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-3,-1,0],[2,3,0]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-3, -1, 0], [2, 3, 0]], [4, 5], 5, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [4, 5], 5, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0036","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ac3a122fe49cc5fd1c8fe1fa00b7a9f11cff6583a0446f3fd8b56a525a8c682","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285036,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-2,0,3],[-2,2,5]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-2, 0, 3], [-2, 2, 5]], [5, 9], 9, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [5, 9], 9, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0037","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ffe761f5b4d37db3b31efea4b20564dd376f51fcf9ee056ca904a69e1487285","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285037,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[-1,1,-5],[-1,1,-3]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[-1, 1, -5], [-1, 1, -3]], [7, 5], 7, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [7, 5], 7, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0038","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a2c4fa83124ee42e55929e8e04f7434c8976778f2014332c1be199fd3fe13a4","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285038,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[0,2,-2],[0,0,2]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[0, 2, -2], [0, 0, 2]], [4, 2], 4, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [4, 2], 4, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0039","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac7ada0a498eb722e46ff0c831b59b8b3c9a5f0d91ba417a39125944d8e6112","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285039,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[1,3,1],[1,-1,-6]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[1, 3, 1], [1, -1, -6]], [5, 8], 8, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [5, 8], 8, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0040","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72e1a1e73e53aa2904f749ef4c1522f17b1beebd1f54b4a55d2484f32500ba95","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285040,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[2,-3,4],[2,-2,-1]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[2, -3, 4], [2, -2, -1]], [9, 5], 9, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [9, 5], 9, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0041","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0cf804e9cc1a5647a15c61e47e43483b9713f6d91abbf282df86597ac13b12b","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285041,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[3,-2,-4],[-2,-3,4]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[3, -2, -4], [-2, -3, 4]], [9, 9], 9, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [9, 9], 9, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0042","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2098b48185bcc5d1110500a4e874aef31554cf2c9841812263766918022a90dc","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285042,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[4,-1,-1],[-1,-4,-4]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[4, -1, -1], [-1, -4, -4]], [6, 9], 9, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [6, 9], 9, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0043","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"558acd2b7b599ee7453ce88e430d3016e2bb2ce8597e3817271a5753a1dfc5c7","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285043,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[5,0,2],[0,-5,1]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[5, 0, 2], [0, -5, 1]], [7, 6], 7, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [7, 6], 7, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0044","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"089e4b335ecb4b8ca7e33599852434d57e6af92c33de759f6fa3a40c4625e30d","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285044,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[6,1,5],[1,-6,6]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[6, 1, 5], [1, -6, 6]], [12, 13], 13, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [12, 13], 13, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0045","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8557e9580aea6b48e2bd2ab39b781955ec79c227a6652753b8e890ecc1b04d7","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285045,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[7,2,-3],[2,-7,-2]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[7, 2, -3], [2, -7, -2]], [12, 11], 12, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [12, 11], 12, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0046","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db1ab8dbe2510a5341ae7f9d6e6cd8ffe4df64be30b8018810e6e34781500044","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285046,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[8,3,0],[-2,-8,3]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[8, 3, 0], [-2, -8, 3]], [11, 13], 13, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [11, 13], 13, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0047","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1931a3abd768dff47722dd3356208f7180479f8793771324a91238fe4e66805","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285047,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[9,-3,3],[-1,-9,-5]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[9, -3, 3], [-1, -9, -5]], [15, 15], 15, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [15, 15], 15, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0048","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b49c43a970ea22d0e8c2cfc0f51fb352e5a63600369bbc01184aa9cb744ae62b","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285048,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[10,-2,-5],[0,-10,0]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[10, -2, -5], [0, -10, 0]], [17, 10], 17, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [17, 10], 17, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0049","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61489e4592e99ff6733586d8bb7ebd0c0e17bf63cd7af74778efc2e92d9094d9","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285049,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[11,-1,-2],[1,-11,5]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[11, -1, -2], [1, -11, 5]], [14, 17], 17, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [14, 17], 17, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0050","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80a7c94029f1b9cfb13044634c1434c4aec8ea8b656b4884a11ec77e16f1fe52","family_id":"qf-mod28-operator-linfinity-norm","seed":202610285050,"semantic_template":"qf-mod28-operator-linfinity-norm/{\"matrix\":[[12,0,1],[2,-12,-3]]}"},"fields":{"title":"The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums","stem":"Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.","answer":{"kind":"expression","value":"([[12, 0, 1], [2, -12, -3]], [13, 17], 17, maximum_absolute_row_sum)","explanation":"Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [13, 17], 17, maximum_absolute_row_sum)."},"skills":["The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0051","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74df86f694417d96429975a7a571fca2cea1bddc70ff82aa3485da080b1a580c","family_id":"qf-mod28-operator-bound-instance","seed":202610285051,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":14,\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"norm_name\":\"l1\",\"vector\":[-4,-2,-3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, -2, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-4, -2, -3], [69, 2], 9, 71, 14, 126, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [69, 2], 9, 71, 14, 126, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0052","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3eb21084d287fbe39a9a75f86c3f1c1739c18aaf8c1055e57780602376510b6","family_id":"qf-mod28-operator-bound-instance","seed":202610285052,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":16,\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"norm_name\":\"linfinity\",\"vector\":[-3,-1,-2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-3, -1, -2], proposed C=16. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-3, -1, -2], [39, -6], 3, 39, 16, 48, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, -2], [39, -6], 3, 39, 16, 48, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0053","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8668066af46a8d5e1404f879ec6491d427351ff273ccdfcf5d0008b59b0a8ef7","family_id":"qf-mod28-operator-bound-instance","seed":202610285053,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":13,\"matrix\":[[-10,-1,1],[0,10,4]],\"norm_name\":\"l1\",\"vector\":[-2,0,-1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-2, 0, -1], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-2, 0, -1], [19, -4], 3, 23, 13, 39, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, -1], [19, -4], 3, 23, 13, 39, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0054","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4cfa2b580cfbc268a2ac7ae6415ecc5dfaa316ea96272eb2203f73ffa053fd4","family_id":"qf-mod28-operator-bound-instance","seed":202610285054,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":14,\"matrix\":[[-9,0,4],[1,9,-4]],\"norm_name\":\"linfinity\",\"vector\":[-1,1,0]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-1, 1, 0], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-1, 1, 0], [9, 8], 1, 9, 14, 14, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, 0], [9, 8], 1, 9, 14, 14, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0055","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfffdade2c0703d6b63a553e12ce8ec5bab4828f0c2c7d4da4b43c1250da688f","family_id":"qf-mod28-operator-bound-instance","seed":202610285055,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[-8,1,-4],[2,8,1]],\"norm_name\":\"l1\",\"vector\":[0,2,1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[0, 2, 1], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [0, 2, 1], [-2, 17], 3, 19, 9, 27, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, 1], [-2, 17], 3, 19, 9, 27, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0056","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7be63cf51ea6405d40c4063a07ce5eece982f0a9909ac3bcb3f0dd17c1177296","family_id":"qf-mod28-operator-bound-instance","seed":202610285056,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":17,\"matrix\":[[-7,2,-1],[-2,7,6]],\"norm_name\":\"linfinity\",\"vector\":[1,-2,2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[1, -2, 2], proposed C=17. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [1, -2, 2], [-13, -4], 2, 13, 17, 34, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, -2, 2], [-13, -4], 2, 13, 17, 34, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0057","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad78beaef7f2635a9ede581f3f03be3e63cca465ec04673784a250fc31871b93","family_id":"qf-mod28-operator-bound-instance","seed":202610285057,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[-6,3,2],[-1,6,-2]],\"norm_name\":\"l1\",\"vector\":[2,-1,3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[2, -1, 3], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [2, -1, 3], [-9, -14], 6, 23, 9, 54, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [2, -1, 3], [-9, -14], 6, 23, 9, 54, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0058","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0df9ef6ddc9b51fdccc8774f3642356839508dcb0a70f8fe8a511db480bd918f","family_id":"qf-mod28-operator-bound-instance","seed":202610285058,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":14,\"matrix\":[[-5,-3,5],[0,5,3]],\"norm_name\":\"linfinity\",\"vector\":[3,0,-3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[3, 0, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [3, 0, -3], [-30, -9], 3, 30, 14, 42, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, 0, -3], [-30, -9], 3, 30, 14, 42, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0059","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"542dc561847d2843012d3b74193227e0374a6678e16f4b09c2ca4e0e6e8977e6","family_id":"qf-mod28-operator-bound-instance","seed":202610285059,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":7,\"matrix\":[[-4,-2,-3],[1,4,-5]],\"norm_name\":\"l1\",\"vector\":[4,1,-2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[4, 1, -2], proposed C=7. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [4, 1, -2], [-12, 18], 7, 30, 7, 49, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [4, 1, -2], [-12, 18], 7, 30, 7, 49, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0060","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a91c4eb12ded91d9c5a164ea06f8895d236cc8d965ff22495e0e8350cd0348d","family_id":"qf-mod28-operator-bound-instance","seed":202610285060,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":5,\"matrix\":[[-3,-1,0],[2,3,0]],\"norm_name\":\"linfinity\",\"vector\":[-4,2,-1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-4, 2, -1], proposed C=5. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-4, 2, -1], [10, -2], 4, 10, 5, 20, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-4, 2, -1], [10, -2], 4, 10, 5, 20, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0061","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f48c49803a8849257dc340de3a62163f27e45747bc74ead0ff919358303ce3a","family_id":"qf-mod28-operator-bound-instance","seed":202610285061,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[-2,0,3],[-2,2,5]],\"norm_name\":\"l1\",\"vector\":[-3,-2,0]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-3, -2, 0], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-3, -2, 0], [6, 2], 5, 8, 9, 45, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-3, -2, 0], [6, 2], 5, 8, 9, 45, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0062","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a38ac6a4c99bb85fa6ded7b7f04f1e6d1eb8b3e847df9c68422ce57522e85df","family_id":"qf-mod28-operator-bound-instance","seed":202610285062,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[-1,1,-5],[-1,1,-3]],\"norm_name\":\"linfinity\",\"vector\":[-2,-1,1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-2, -1, 1], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-2, -1, 1], [-4, -2], 2, 4, 9, 18, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-2, -1, 1], [-4, -2], 2, 4, 9, 18, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0063","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228f0190683d4b1b3db53845650fe929b853a2f4efbe628a3311e6ad504a72a5","family_id":"qf-mod28-operator-bound-instance","seed":202610285063,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":3,\"matrix\":[[0,2,-2],[0,0,2]],\"norm_name\":\"l1\",\"vector\":[-1,0,2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-1, 0, 2], proposed C=3. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-1, 0, 2], [-4, 4], 3, 8, 3, 9, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-1, 0, 2], [-4, 4], 3, 8, 3, 9, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0064","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f3cf74010649b6061a83b510c761abe74d45d4de8a67b82dcc4fc94dbba7d8","family_id":"qf-mod28-operator-bound-instance","seed":202610285064,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[1,3,1],[1,-1,-6]],\"norm_name\":\"linfinity\",\"vector\":[0,1,3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[0, 1, 3], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [0, 1, 3], [6, -19], 3, 19, 9, 27, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [0, 1, 3], [6, -19], 3, 19, 9, 27, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0065","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fbf6207daf1136a6cf3d0c1d043da490084c1ef7fe27de6c4ee5ab1bf79416","family_id":"qf-mod28-operator-bound-instance","seed":202610285065,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":7,\"matrix\":[[2,-3,4],[2,-2,-1]],\"norm_name\":\"l1\",\"vector\":[1,2,-3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[1, 2, -3], proposed C=7. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [1, 2, -3], [-16, 1], 6, 17, 7, 42, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [1, 2, -3], [-16, 1], 6, 17, 7, 42, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0066","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92f430548b1d490a2969cce74abeb4529db6b4e500320d0882c3e08f82d5e49","family_id":"qf-mod28-operator-bound-instance","seed":202610285066,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[3,-2,-4],[-2,-3,4]],\"norm_name\":\"linfinity\",\"vector\":[2,-2,-2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[2, -2, -2], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [2, -2, -2], [18, -6], 2, 18, 9, 18, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [2, -2, -2], [18, -6], 2, 18, 9, 18, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0067","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad35cc689b7356cba66ea1932f3dff1cc159c6bf9c61865028a91ba9a03427a6","family_id":"qf-mod28-operator-bound-instance","seed":202610285067,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":4,\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"norm_name\":\"l1\",\"vector\":[3,-1,-1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[3, -1, -1], proposed C=4. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [3, -1, -1], [14, 5], 5, 19, 4, 20, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [3, -1, -1], [14, 5], 5, 19, 4, 20, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0068","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d154cfec52f1dc819739864f67b3f759cd4c2492c8f983b3e85216e166e88cd7","family_id":"qf-mod28-operator-bound-instance","seed":202610285068,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":9,\"matrix\":[[5,0,2],[0,-5,1]],\"norm_name\":\"linfinity\",\"vector\":[4,0,0]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[4, 0, 0], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [4, 0, 0], [20, 0], 4, 20, 9, 36, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [4, 0, 0], [20, 0], 4, 20, 9, 36, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0069","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba82be2b801464f7652836419cf6f6b7c49ffb06155febee6e6a4357f7e582bd","family_id":"qf-mod28-operator-bound-instance","seed":202610285069,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":11,\"matrix\":[[6,1,5],[1,-6,6]],\"norm_name\":\"l1\",\"vector\":[-4,1,1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, 1, 1], proposed C=11. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-4, 1, 1], [-18, -4], 6, 22, 11, 66, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-4, 1, 1], [-18, -4], 6, 22, 11, 66, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0070","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a137746c1636e11af87de12df799535b47cd36712547b2e5557690b5c56909f","family_id":"qf-mod28-operator-bound-instance","seed":202610285070,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":13,\"matrix\":[[7,2,-3],[2,-7,-2]],\"norm_name\":\"linfinity\",\"vector\":[-3,2,2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-3, 2, 2], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-3, 2, 2], [-23, -24], 3, 24, 13, 39, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, 2, 2], [-23, -24], 3, 24, 13, 39, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0071","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e061aa9bb16927c08112b1fcd871fb2d001394f595b9aed668a5c1c1aa0269","family_id":"qf-mod28-operator-bound-instance","seed":202610285071,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":10,\"matrix\":[[8,3,0],[-2,-8,3]],\"norm_name\":\"l1\",\"vector\":[-2,-2,3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-2, -2, 3], proposed C=10. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [-2, -2, 3], [-22, 29], 7, 51, 10, 70, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-2, -2, 3], [-22, 29], 7, 51, 10, 70, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0072","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd8a0ab3457da608d6f958cacc7fc2f68dc572baa4c5b77edd6f29933a271d51","family_id":"qf-mod28-operator-bound-instance","seed":202610285072,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":15,\"matrix\":[[9,-3,3],[-1,-9,-5]],\"norm_name\":\"linfinity\",\"vector\":[-1,-1,-3]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-1, -1, -3], proposed C=15. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [-1, -1, -3], [-15, 25], 3, 25, 15, 45, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, -1, -3], [-15, 25], 3, 25, 15, 45, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0073","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd1a43ea5b91c66bfae2ac13f9d0540a03828ee5c49a2eebdaa73be1cef964ea","family_id":"qf-mod28-operator-bound-instance","seed":202610285073,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":13,\"matrix\":[[10,-2,-5],[0,-10,0]],\"norm_name\":\"l1\",\"vector\":[0,0,-2]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[0, 0, -2], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [0, 0, -2], [10, 0], 2, 10, 13, 26, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [0, 0, -2], [10, 0], 2, 10, 13, 26, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0074","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f60aedb85f9933f263256341f1448a6af11eba42e700f8ab518a6442dfbacc7","family_id":"qf-mod28-operator-bound-instance","seed":202610285074,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":19,\"matrix\":[[11,-1,-2],[1,-11,5]],\"norm_name\":\"linfinity\",\"vector\":[1,1,-1]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[1, 1, -1], proposed C=19. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(linfinity, [1, 1, -1], [12, -15], 1, 15, 19, 19, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 1, -1], [12, -15], 1, 15, 19, 19, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0075","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3d835688ba64d0b7091124b889fa47be67b30a4e3d134ecc2bfbb32b3add334","family_id":"qf-mod28-operator-bound-instance","seed":202610285075,"semantic_template":"qf-mod28-operator-bound-instance/{\"C\":13,\"matrix\":[[12,0,1],[2,-12,-3]],\"norm_name\":\"l1\",\"vector\":[2,2,0]}"},"fields":{"title":"One Passing Vector Does Not Certify a Uniform Bound","stem":"Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[2, 2, 0], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).","answer":{"kind":"expression","value":"(l1, [2, 2, 0], [24, -20], 4, 44, 13, 52, Yes, instance_only)","explanation":"Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [2, 2, 0], [24, -20], 4, 44, 13, 52, Yes, instance_only)."},"skills":["One Passing Vector Does Not Certify a Uniform 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-20261028-r05-bounded-linear-operator-0076","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b9348f58d6f9ac2062d37f304a909236fd8ddf4974ead5b0839fafb58a9e8d3","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285076,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":14,\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 15, 14, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 15, 14, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0077","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cd533662fd0f46d2dcc0b565bd800617cd979dd9aa5ee402c3bc5d2329b2e17","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285077,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":16,\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=16. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 15, 16, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 16, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0078","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93c545ff887fafa4175406903723d9fd85ef07d89bd074883162be55a09ab5cb","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285078,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":13,\"matrix\":[[-10,-1,1],[0,10,4]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 11, 13, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 13, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0079","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e99cc4a1f0a535185470589df95948f03199e5fbfe35dfde3ce128a6a5b2a16","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285079,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":14,\"matrix\":[[-9,0,4],[1,9,-4]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 14, 14, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0080","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64235939c79bf1044f08d04a5d80cab96c07bd489f7b783f449542092a160841","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285080,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[-8,1,-4],[2,8,1]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 10, 9, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 10, 9, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0081","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27d7f772db759d8737f151f228f888b509fc262d02df1018f640cd7cbfdcb7e0","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285081,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":17,\"matrix\":[[-7,2,-1],[-2,7,6]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=17. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 15, 17, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 17, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0082","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f9130e17e2e85e9e8b7198c87f13d14cb0cbd0561739ec9a6d267dd35195ab","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285082,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[-6,3,2],[-1,6,-2]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 9, 9, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 9, 9, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0083","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaa89ba28e2d30b2135705670aaa7de241c2a1e5b12a44e57f5127705d2e28e7","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285083,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":14,\"matrix\":[[-5,-3,5],[0,5,3]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 13, 14, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 13, 14, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0084","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac8b8de52c54b7d36575dbf6c915e44b74f477ff37d0b9da2859a01e8b8fe47","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285084,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":7,\"matrix\":[[-4,-2,-3],[1,4,-5]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=7. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 8, 7, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 8, 7, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0085","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c77760f588934e0a46fc0dd108696607046f53702101c00dd59efcacf05eb80f","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285085,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":5,\"matrix\":[[-3,-1,0],[2,3,0]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=5. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 5, 5, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 5, 5, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0086","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20e4116a3d433a4fda01b5a14750b3a3f22f55cf376fdb5a199f6a9b7a5396f4","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285086,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[-2,0,3],[-2,2,5]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 8, 9, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 8, 9, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0087","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"827db40caac0124cca3819d8953d3ceb3a5edff8fad4c80ebf20bfaa7fbb08d8","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285087,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[-1,1,-5],[-1,1,-3]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 7, 9, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 9, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0088","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac8c94a4c6ed5692ff742ec0a1cfdc96bddebcb5c9127ce16087386ab13e386a","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285088,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":3,\"matrix\":[[0,2,-2],[0,0,2]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=3. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 4, 3, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 4, 3, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0089","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e74320b09ca7b414de0f84eda26c8f54da94777f0742a7313a27d11af4627e26","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285089,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[1,3,1],[1,-1,-6]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 8, 9, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, 9, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0090","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ae724f43e59a89770381dcd7d66b7af299cc1eb43f85b3e0d02bf105cf68bc9","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285090,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":7,\"matrix\":[[2,-3,4],[2,-2,-1]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=7. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 5, 7, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 5, 7, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0091","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b35d4392155d870142b44e0d70e0135d72000166bb16934b5e67eea66f2e9e67","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285091,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[3,-2,-4],[-2,-3,4]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 9, 9, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 9, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0092","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27c1a4dee9f3033410649d7ffe7ad4db885f18fa1e31d06e5df41ee4211e32cb","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285092,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":4,\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=4. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 5, 4, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 5, 4, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0093","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b70d3619d59c2825c5919c6b41894e9906874d6579f09d05645789c559800c50","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285093,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":9,\"matrix\":[[5,0,2],[0,-5,1]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 7, 9, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 9, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0094","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c83b1b974183d3ec4e6e2add70ccad2e2e8dd687b208df8f55c525773cc7a91","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285094,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":11,\"matrix\":[[6,1,5],[1,-6,6]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=11. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 11, 11, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 11, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0095","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e81a3ec041a94b738e5745579efc8898507ffef6eb6445f0b3a982b27075f9d","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285095,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":13,\"matrix\":[[7,2,-3],[2,-7,-2]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 12, 13, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, 13, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0096","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a8f4b8a045f3d9f9af322e94e186e9cc8c536922724f5b4d115f168aa50434d","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285096,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":10,\"matrix\":[[8,3,0],[-2,-8,3]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=10. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 11, 10, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 10, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0097","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb63ff689039c4548e08b6a5571e5e37d1aca3d236a113c2a0a24c5be92e34ca","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285097,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":15,\"matrix\":[[9,-3,3],[-1,-9,-5]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=15. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 15, 15, Yes, Yes, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 15, Yes, Yes, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0098","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7b3ea50fd16759e72ca12a2631a1fbe845627013d05b6eb334b3d496aaf18eb","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285098,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":13,\"matrix\":[[10,-2,-5],[0,-10,0]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 12, 13, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 12, 13, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0099","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9205dac4b23842e8b7c39efeaa21acd9526cc1a7688ee793ce7f183f0d84db16","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285099,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":19,\"matrix\":[[11,-1,-2],[1,-11,5]],\"norm_name\":\"linfinity\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=19. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(linfinity, 17, 19, Yes, No, uniform_bound_certificate)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 17, 19, Yes, No, uniform_bound_certificate)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r05-bounded-linear-operator-0100","batch_id":"batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b29fd4c6abf3e2d50bd8f26f7eedfeab5817f1c1fdfa0112dc019dabe545633","family_id":"qf-mod28-operator-bounded-certificate","seed":202610285100,"semantic_template":"qf-mod28-operator-bounded-certificate/{\"C\":13,\"matrix\":[[12,0,1],[2,-12,-3]],\"norm_name\":\"l1\"}"},"fields":{"title":"A Usable Uniform Bound May Be Nonoptimal","stem":"Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.","answer":{"kind":"expression","value":"(l1, 14, 13, No, No, bound_too_small)","explanation":"Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 14, 13, No, No, bound_too_small)."},"skills":["A Usable Uniform Bound May Be Nonoptimal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}