{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0001","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07cae5dd38b935bf8530986619fe7c2d10c5b7252f4b90c76cce8c228bdebb36","family_id":"qf-str22-metric-table-qualification","seed":202610226001,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0002","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ba518144806ba2b0eff678372d7957aa618a18ac835196ef14692b05ecce7e4","family_id":"qf-str22-metric-table-qualification","seed":202610226002,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0003","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7cc0b92aaba3f3e874b7d7b24fc3862029499678cf8bd967ca4bc4bc3f1b56d","family_id":"qf-str22-metric-table-qualification","seed":202610226003,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0004","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57d37f400f1a58322f1450b4e90240898e68f9c92fcc20e7087814102c4c6ac1","family_id":"qf-str22-metric-table-qualification","seed":202610226004,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0005","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee436ad57b1bb025b2e14319b4efa8476a97fff04b9975b00c2752bbf3965ab","family_id":"qf-str22-metric-table-qualification","seed":202610226005,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0006","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63afcf21674e485c41052996cab586209e461b21890157ccbdcb6d72fbf4fc12","family_id":"qf-str22-metric-table-qualification","seed":202610226006,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0007","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8b5c6db10319c87afae5d1c165f433b7b133171f84bca08bdb29edb5cc699c3","family_id":"qf-str22-metric-table-qualification","seed":202610226007,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0008","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c265945875df8d576cdc5b0e4342dc4e7606e742b3cb126ac52f081dfd033a1","family_id":"qf-str22-metric-table-qualification","seed":202610226008,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0009","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be99aeaa4f782b4c03c3af3ebf956363077b8dcab4190280727040fa76b118d2","family_id":"qf-str22-metric-table-qualification","seed":202610226009,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0010","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b7da766a48bd65fc3215a502f01ce73926f64bff66297b8b62323208fd0a8b","family_id":"qf-str22-metric-table-qualification","seed":202610226010,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0011","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c295a4993b128915560fd69bb25952541622ce530447407e093218d7a2ea2829","family_id":"qf-str22-metric-table-qualification","seed":202610226011,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0012","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8df8df9ff2fbbd0603660e6da0454e53811fcc7fc52cdf10059146d7ca69d81c","family_id":"qf-str22-metric-table-qualification","seed":202610226012,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0013","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1931f748563921e02b6faa64ee8a9c608ee3d19d3ff8f0904dc3a19fd95cafda","family_id":"qf-str22-metric-table-qualification","seed":202610226013,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0014","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e90b36f6af1670e0b571bb6f7acc637be2392364e0d8371913ff27b867a50268","family_id":"qf-str22-metric-table-qualification","seed":202610226014,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0015","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e12dc1f01abd9b1523321d61b901d5f3fc7d194dbdcc5412a4ab9480ead2589b","family_id":"qf-str22-metric-table-qualification","seed":202610226015,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0016","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e054c25840138654ab07362b6ad9a0381011bafa5732bab2219e75edda2c1ca3","family_id":"qf-str22-metric-table-qualification","seed":202610226016,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0017","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20373f318916e159e11eba4baf38ff8f9a32887544e43cd5cf78d5580afa0136","family_id":"qf-str22-metric-table-qualification","seed":202610226017,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0018","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaab2d140d8da95e95c1af940e77739c67bd200a7a0abb1bd9adc063e1b2ae20","family_id":"qf-str22-metric-table-qualification","seed":202610226018,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0019","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"443e0b6e770405fec70d78fbde65df01fa91714487cfe193da335fcb8afd7707","family_id":"qf-str22-metric-table-qualification","seed":202610226019,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0020","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d1a1b2470402221473812c89961e6fe250a85f1125504b26cef3fd730aa3b73","family_id":"qf-str22-metric-table-qualification","seed":202610226020,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0021","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7672684a3c1c313b65b9763f3fa297ef0162671234b2ff666eeef4128c7906e3","family_id":"qf-str22-metric-table-qualification","seed":202610226021,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0022","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc83bab10949585f336481063c16f551d6b2c7cdd0b97dd301db0ba3e20c2a7","family_id":"qf-str22-metric-table-qualification","seed":202610226022,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0023","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"657e506727d189c161ba3f3ad66dc1498581061b53de57a5e8b484a895f0328e","family_id":"qf-str22-metric-table-qualification","seed":202610226023,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0024","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc535434cf74a0a900170fdae49e0a4859b7f7751d9f89673fbc1762a16abb45","family_id":"qf-str22-metric-table-qualification","seed":202610226024,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0025","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1703bd9d0dd92b5f448e25b42ed1fb29e4fa3228697973bf8da5091e96bc64a1","family_id":"qf-str22-metric-table-qualification","seed":202610226025,"semantic_template":"qf-str22-metric-table-qualification/{\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4}"},"fields":{"title":"Nonnegativity and Separation Are Independent Table Checks","stem":"The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).","answer":{"kind":"expression","value":"(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes)","explanation":"Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes)."},"skills":["Nonnegativity and Separation Are Independent Table Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0026","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb668ce2a286a7594b87e73f0fa621b8325fc2e586c48c400b8b207996f1556","family_id":"qf-str22-metric-symmetry-audit","seed":202610226026,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0027","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"298ebf08c22e6f516ae2aafe4566ba9ca93e1f56e114a281c3218bc1097d58bc","family_id":"qf-str22-metric-symmetry-audit","seed":202610226027,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0028","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65fcde68eaae2066636859b35ab8065afdae9fc2e82ef4ce359724613a52ebf2","family_id":"qf-str22-metric-symmetry-audit","seed":202610226028,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0029","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72fe331ab4b6e32b8f339337fe8dede9d59c25d0167f68ce87b7c1eb9c29db2c","family_id":"qf-str22-metric-symmetry-audit","seed":202610226029,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0030","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0710a4f5c8dc6af382de78811632ba6a3245a92c13b754d14a24a05ddfc8a26f","family_id":"qf-str22-metric-symmetry-audit","seed":202610226030,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0031","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e237dddc6a9d682a65cd75f0b11c94374dc398406a0bbea0cd72e1825cdfa71","family_id":"qf-str22-metric-symmetry-audit","seed":202610226031,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0032","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b267932aed3446f088388c9c2cc4b44be86bf46c11b95bf3d3c64d2feb070ea4","family_id":"qf-str22-metric-symmetry-audit","seed":202610226032,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0033","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b39d89f67d3c19a542d00dca261b5dfc9db1480f4e6094d65781dc6c0225ac39","family_id":"qf-str22-metric-symmetry-audit","seed":202610226033,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0034","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68f313d5e59506d5bfc8eb820450682599462598a75b1d75149c1e4550c0142d","family_id":"qf-str22-metric-symmetry-audit","seed":202610226034,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0035","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a12a2fa3748508e98f638704afa0553afcc369b44377509039b5e058e05b115","family_id":"qf-str22-metric-symmetry-audit","seed":202610226035,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0036","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1f3e465bbe46de9bd3c19e705a6a375b6325b7ef17cb0be8715fed7770881c","family_id":"qf-str22-metric-symmetry-audit","seed":202610226036,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0037","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30719735cdb9652176898f3577ae2e77f2e74fd7cddab792f2818a50f8d0c52b","family_id":"qf-str22-metric-symmetry-audit","seed":202610226037,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0038","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06049b04503afa7b70b86b9213427d3095f8924b3cf3ec4142498637ff7ededc","family_id":"qf-str22-metric-symmetry-audit","seed":202610226038,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0039","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff6c85a667b04b4431cdfdbbe349b6f7504558331b4e5370c4411f94537995d6","family_id":"qf-str22-metric-symmetry-audit","seed":202610226039,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0040","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e56e0da85508c9ac27e3e2247666d6b0cfda24bcb8a674cfc7a20d9967c8f47","family_id":"qf-str22-metric-symmetry-audit","seed":202610226040,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0041","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02a272568ed8a0172f38adfb1b9a5e38af1e138f09453935bdb40f04cd9a3743","family_id":"qf-str22-metric-symmetry-audit","seed":202610226041,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0042","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac5ef07f3b8d1b0a0cd372955bd58f47bb85e34ab7fed5ffb5b4b5a6536de287","family_id":"qf-str22-metric-symmetry-audit","seed":202610226042,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0043","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343972bf9d7783d6732d6c578c804f2927c6922d39e7414b6b986c255db5c11e","family_id":"qf-str22-metric-symmetry-audit","seed":202610226043,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0044","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acd95a4ce111c8a7e8f7adac40642f1a67097b8e1e43d150b25cab36103a424d","family_id":"qf-str22-metric-symmetry-audit","seed":202610226044,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0045","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b55a7eedaae3a56fef0cd05d55acdd7f08d30f262a201ff14a4c314e31972afb","family_id":"qf-str22-metric-symmetry-audit","seed":202610226045,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0046","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43e8eb356790854478ce62d5a93f95761d3d292d4c8ab607b44d8288ebe72ede","family_id":"qf-str22-metric-symmetry-audit","seed":202610226046,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0047","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db7918401b8ed0844a680bb0f565cf6f163e5657d209b81c00e260b4bcae0fbf","family_id":"qf-str22-metric-symmetry-audit","seed":202610226047,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0048","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd37382b628f65ae53947440b978a915df3316fa163354774c69c1fb979e1ef9","family_id":"qf-str22-metric-symmetry-audit","seed":202610226048,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0049","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c147c59903906a2a93eae461ec2aa6743ab085416d1a45ce3d867fe72a376b7d","family_id":"qf-str22-metric-symmetry-audit","seed":202610226049,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0050","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe38c6735d44aa77551c9e89ab1bdf6d3e119447b4a48c3b8b668fd236d36d3","family_id":"qf-str22-metric-symmetry-audit","seed":202610226050,"semantic_template":"qf-str22-metric-symmetry-audit/{\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4}"},"fields":{"title":"Compare Both Directions of Every Distance Pair","stem":"The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).","answer":{"kind":"expression","value":"(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes)","explanation":"Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes)."},"skills":["Compare Both Directions of Every Distance Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0051","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1ebb32acc0e942f174bc0c4b912975a24097a5dc6d1b76721963ec886a77a2d","family_id":"qf-str22-metric-triangle-audit","seed":202610226051,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0052","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"076263a558e3e7ff3fb3ba06a4f66d21e08aefd62ede3a241ea71c7375463087","family_id":"qf-str22-metric-triangle-audit","seed":202610226052,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0053","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e63e48dbbce1be9ef02ce695aee0b4f35c6d047f764ab09be1e3df1ed1b12ff3","family_id":"qf-str22-metric-triangle-audit","seed":202610226053,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0054","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f3a2675336dd1129b5baf9a33415af6a92cbe6ad7b46db9186cb3689b5c4c6","family_id":"qf-str22-metric-triangle-audit","seed":202610226054,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0055","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55656df380a424a8fdac10f883753a8b4d407124d1121822fbaec7c246ef91c3","family_id":"qf-str22-metric-triangle-audit","seed":202610226055,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0056","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a46f6d44934567a0754ec8e8f4176d0c8f6197942ee8b6a239758ad762d4c8b","family_id":"qf-str22-metric-triangle-audit","seed":202610226056,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0057","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb8dbb62d04013b192d6b7b72156789a36b065c89f0efd87aafeb1795259116","family_id":"qf-str22-metric-triangle-audit","seed":202610226057,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0058","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d1b6d917e1e5d0bf2626a8548339524628e480fe41dfcca5c1f0d0985b5e26f","family_id":"qf-str22-metric-triangle-audit","seed":202610226058,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0059","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c94d19b4ceaf5e6ac5bc83321260ed9f01ae9d67b593a5aaa4ad559b639020dc","family_id":"qf-str22-metric-triangle-audit","seed":202610226059,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0060","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48dc29de772dac2970a71d6394ee89135a46328d0fbd8062fe0a1d351c3ecbe8","family_id":"qf-str22-metric-triangle-audit","seed":202610226060,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0061","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c776397258e54e2fc904c7e237bf0268dc4aef5934e5369a631c46825964a2","family_id":"qf-str22-metric-triangle-audit","seed":202610226061,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0062","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ab8a3744653262c6dd7127f74e300bec50cfca60994d0f2b58548aa7d9ef30","family_id":"qf-str22-metric-triangle-audit","seed":202610226062,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0063","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d37c3163f0985f9058073fef8f53e716e4abf35a9d35b48852093949bcdc8036","family_id":"qf-str22-metric-triangle-audit","seed":202610226063,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0064","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59db442b4b2db84f9225c65dc188e25514af6a65ec986b67e860400df3cd780a","family_id":"qf-str22-metric-triangle-audit","seed":202610226064,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0065","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"162d452d114caac14f80077f80bae18e4f1e95f33fb6bfef0ac784ca238d739f","family_id":"qf-str22-metric-triangle-audit","seed":202610226065,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0066","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"131e9c41d614cd71510f94f18bb357b7a0f260f44d13604661a47b0be1d7ea04","family_id":"qf-str22-metric-triangle-audit","seed":202610226066,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0067","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85243d72682aca344843efd12cf502a33ef9394a015d8912da6a5c5fc7ef1bb3","family_id":"qf-str22-metric-triangle-audit","seed":202610226067,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0068","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d32efb3f59f41021f23f55fd85fe1e1d20f7b24ae35d3242cf13f0081ef5d004","family_id":"qf-str22-metric-triangle-audit","seed":202610226068,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0069","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c1aa3b2a889a100c8d48a417badf6e47086fcf20f1c935eed813d7374fd50f5","family_id":"qf-str22-metric-triangle-audit","seed":202610226069,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0070","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ea1ff251d73ec55434d631a754fa1b213df25281c032fcf8a48b77668fffb19","family_id":"qf-str22-metric-triangle-audit","seed":202610226070,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0071","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7911497f28506e74a51a84e737497561b966db5bc60e37c4e687ed75d921559f","family_id":"qf-str22-metric-triangle-audit","seed":202610226071,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0072","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61479bd30a7f078d063eb4b90297660591affc3e46bb041d4c933ec10181281e","family_id":"qf-str22-metric-triangle-audit","seed":202610226072,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0073","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"104c7066cf77971c9e4444cc791e40619312378ea02263183b5086a296c9c7c8","family_id":"qf-str22-metric-triangle-audit","seed":202610226073,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0074","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0017c5c6e8270ee8681be6245b88d73b5eac629dfccb7b178a3a248a7ad6474","family_id":"qf-str22-metric-triangle-audit","seed":202610226074,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0075","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7320bbb26c09b441227a3fa2a2d39191896b99cad56b0c8848465016c3fd3741","family_id":"qf-str22-metric-triangle-audit","seed":202610226075,"semantic_template":"qf-str22-metric-triangle-audit/{\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4}"},"fields":{"title":"Every Ordered Triangle Inequality in the Finite Table","stem":"The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.","answer":{"kind":"expression","value":"(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], [], Yes)","explanation":"Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], [], Yes)."},"skills":["Every Ordered Triangle Inequality in the Finite Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0076","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"624bcb2db50f837a96f118768aa685aa29ec974e8b5645f11cadbb1cb20666a0","family_id":"qf-str22-metric-full-metric-audit","seed":202610226076,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,1,3,6],[1,0,2,5],[3,2,0,3],[6,5,3,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0077","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63c32e725852251e6d28ba67b5cb477542032003f20ae055961247ea24ddbc0f","family_id":"qf-str22-metric-full-metric-audit","seed":202610226077,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0078","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11f7206efe90844610a7d1bd77ad1844af8630173f12b0d6d3aa25b008a84019","family_id":"qf-str22-metric-full-metric-audit","seed":202610226078,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0079","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0653dc390e5e927e113c3f7b8b728a21564d46371c989401e409c030ac273f4","family_id":"qf-str22-metric-full-metric-audit","seed":202610226079,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,4,9,15],[4,0,5,11],[9,5,0,6],[15,11,6,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0080","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c9c7406171c3e9363f96d6b7a7c78823bd2fef9c9e8edb8f4ed9254dd6fc8ca","family_id":"qf-str22-metric-full-metric-audit","seed":202610226080,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,5,11,18],[5,0,6,13],[11,6,0,7],[18,13,7,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0081","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a2fbf050acbdce03e73779fc8f9e199b8aad68f23731231c79ec6615ea2fd18","family_id":"qf-str22-metric-full-metric-audit","seed":202610226081,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,6,13,21],[6,0,7,15],[13,7,0,8],[21,15,8,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0082","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"115b34178d47f88e3d6f5f51e1127a5bcc4a6f35c25e025f75e04f5ffa4106bf","family_id":"qf-str22-metric-full-metric-audit","seed":202610226082,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,7,15,24],[7,0,8,17],[15,8,0,9],[24,17,9,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0083","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8b38182e8209083d86994c8eada0d7dd14380af723498c9f9d66f429e2915fd","family_id":"qf-str22-metric-full-metric-audit","seed":202610226083,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,8,17,27],[8,0,9,19],[17,9,0,10],[27,19,10,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0084","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada83589a3dc285e320c20ce3a7e1e13a1ad45af4e3ec68cb11a92bd8e5e2977","family_id":"qf-str22-metric-full-metric-audit","seed":202610226084,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,9,19,30],[9,0,10,21],[19,10,0,11],[30,21,11,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0085","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad670bcb13b639ce5c7ba5a2bb8f88c5c4520cc5794076e4ea25ab7d39eb6c54","family_id":"qf-str22-metric-full-metric-audit","seed":202610226085,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,10,21,33],[10,0,11,23],[21,11,0,12],[33,23,12,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0086","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6fcbb6df8cd4826a782798e039affd3403ef0608c3f9a88e5ff2b52fd6775f3","family_id":"qf-str22-metric-full-metric-audit","seed":202610226086,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,11,23,36],[11,0,12,25],[23,12,0,13],[36,25,13,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0087","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"600c03f48b77631e64cc71c4bd408062d26488cbc1f3f7f10ef5e308c7a6aa2c","family_id":"qf-str22-metric-full-metric-audit","seed":202610226087,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,12,25,39],[12,0,13,27],[25,13,0,14],[39,27,14,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0088","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"524a31d95b283da605f1a31faefc8245ae45ecd581e4cd783aff070a35539685","family_id":"qf-str22-metric-full-metric-audit","seed":202610226088,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,13,27,42],[13,0,14,29],[27,14,0,15],[42,29,15,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0089","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b0e3f91b787fce6f1746e8e4f0ed2c622e9be80487512139e30c8c96cd5600","family_id":"qf-str22-metric-full-metric-audit","seed":202610226089,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,14,29,45],[14,0,15,31],[29,15,0,16],[45,31,16,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0090","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c663b4596062fd2a1157edca2a83ec4e37b61367ef954d8371d9bcef470aef","family_id":"qf-str22-metric-full-metric-audit","seed":202610226090,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,15,31,48],[15,0,16,33],[31,16,0,17],[48,33,17,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0091","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cf3fc9b345ae3d0bb839c1d0e25f28fd809bfb01c1ef6cb0d742830d739e78d","family_id":"qf-str22-metric-full-metric-audit","seed":202610226091,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,16,33,51],[16,0,17,35],[33,17,0,18],[51,35,18,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0092","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffcf2bbf079aadeda9e99de3e3021cd76085c55bdedcde39827a1b36c2231328","family_id":"qf-str22-metric-full-metric-audit","seed":202610226092,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,17,35,54],[17,0,18,37],[35,18,0,19],[54,37,19,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0093","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a97f67a9e2c5de9878f960eabcb96fcfc3e7b42e87dc238f5fd6aedbcd0b8e","family_id":"qf-str22-metric-full-metric-audit","seed":202610226093,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,18,37,57],[18,0,19,39],[37,19,0,20],[57,39,20,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0094","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"847eadb4856ed89894be70ed15c7f4040be9cc8c9b705ec80ec3390cd0f536dc","family_id":"qf-str22-metric-full-metric-audit","seed":202610226094,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,19,39,60],[19,0,20,41],[39,20,0,21],[60,41,21,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0095","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27322232e169fa4f50e87cafedbe0847a88fc66b5690e188962e23709fdac7a7","family_id":"qf-str22-metric-full-metric-audit","seed":202610226095,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,20,41,63],[20,0,21,43],[41,21,0,22],[63,43,22,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0096","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a412fb1635697bd82c174ee6f012ec61169f0db48caa2f88f64a4774969b227e","family_id":"qf-str22-metric-full-metric-audit","seed":202610226096,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,21,43,66],[21,0,22,45],[43,22,0,23],[66,45,23,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0097","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad12e49627dae7706a0c56c8ef0d78ea51cae9c61d164f1a207a3e555d36287a","family_id":"qf-str22-metric-full-metric-audit","seed":202610226097,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,22,45,69],[22,0,23,47],[45,23,0,24],[69,47,24,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0098","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930d86a0862bfd536420e32bd333853f4e29b74aaed05e90429d15de4a3b8402","family_id":"qf-str22-metric-full-metric-audit","seed":202610226098,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,23,47,72],[23,0,24,49],[47,24,0,25],[72,49,25,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0099","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aea1dc9eca14f8e4e18bcc8fc1604d2c6708230288d9f0261bfb230c3d899fe","family_id":"qf-str22-metric-full-metric-audit","seed":202610226099,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,24,49,75],[24,0,25,51],[49,25,0,26],[75,51,26,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r06-metric-space-0100","batch_id":"batch-20261013-advance-20261022-r06-metric-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e55452b3bd6ad2b519214d2a5d8b444599934068d999e2510c6284f6658138b8","family_id":"qf-str22-metric-full-metric-audit","seed":202610226100,"semantic_template":"qf-str22-metric-full-metric-audit/{\"distances\":[[0,25,51,78],[25,0,26,53],[51,26,0,27],[78,53,27,0]],\"n\":4}"},"fields":{"title":"All Metric Axioms on a Complete Finite Distance Table","stem":"The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).","answer":{"kind":"expression","value":"(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes, Yes, [], Yes)","explanation":"Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes, Yes, [], Yes)."},"skills":["All Metric Axioms on a Complete Finite Distance Table"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}