{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0001","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"103b7c822cdf35454d5e5bb75897f71fcb092efcdd71eab39aad253371b4ebec","family_id":"qf-mod47-contract-metric-qualification","seed":202611161001,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1/2, Yes], [0, 2, 1, 3/2, Yes], [1, 0, 0, 1/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [2, 0, 1, 3/2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1/2, Yes], [0, 2, 1, 3/2, Yes], [1, 0, 0, 1/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [2, 0, 1, 3/2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0002","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e795f7fe597a9cb1164d7611c6ceb8fbfc693964ed105d294c514dee10a0ef","family_id":"qf-mod47-contract-metric-qualification","seed":202611161002,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1, Yes], [0, 2, 2, 3, Yes], [1, 0, 0, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 2, 2, Yes], [2, 0, 2, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1, Yes], [0, 2, 2, 3, Yes], [1, 0, 0, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 2, 2, Yes], [2, 0, 2, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0003","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e8eb54908db7ac0093e3e141e5459ccb152874119565d5af6646def2d7c196","family_id":"qf-mod47-contract-metric-qualification","seed":202611161003,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3/2, Yes], [0, 2, 3, 9/2, Yes], [1, 0, 0, 3/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 3, 3, Yes], [2, 0, 3, 9/2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3/2, Yes], [0, 2, 3, 9/2, Yes], [1, 0, 0, 3/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 3, 3, Yes], [2, 0, 3, 9/2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0004","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"073e2b9fc9ebcb9efe2605debc370766b49b2645883f73a4ec4d71ca4f4a13df","family_id":"qf-mod47-contract-metric-qualification","seed":202611161004,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 2, Yes], [0, 2, 4, 6, Yes], [1, 0, 0, 2, Yes], [1, 1, 0, 0, Yes], [1, 2, 4, 4, Yes], [2, 0, 4, 6, Yes], [2, 1, 4, 4, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 2, Yes], [0, 2, 4, 6, Yes], [1, 0, 0, 2, Yes], [1, 1, 0, 0, Yes], [1, 2, 4, 4, Yes], [2, 0, 4, 6, Yes], [2, 1, 4, 4, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0005","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8b5fcace464833aca5d74a612fee6744bb831d9cb7562bb2f555c73ccc0b6a5","family_id":"qf-mod47-contract-metric-qualification","seed":202611161005,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5/2, Yes], [0, 2, 5, 15/2, Yes], [1, 0, 0, 5/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 5, 5, Yes], [2, 0, 5, 15/2, Yes], [2, 1, 5, 5, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5/2, Yes], [0, 2, 5, 15/2, Yes], [1, 0, 0, 5/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 5, 5, Yes], [2, 0, 5, 15/2, Yes], [2, 1, 5, 5, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0006","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af55fb2e5dec68b029d53cbafcee270783aac379feaca142a85bf45791b399cc","family_id":"qf-mod47-contract-metric-qualification","seed":202611161006,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3, Yes], [0, 2, 6, 9, Yes], [1, 0, 0, 3, Yes], [1, 1, 0, 0, Yes], [1, 2, 6, 6, Yes], [2, 0, 6, 9, Yes], [2, 1, 6, 6, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3, Yes], [0, 2, 6, 9, Yes], [1, 0, 0, 3, Yes], [1, 1, 0, 0, Yes], [1, 2, 6, 6, Yes], [2, 0, 6, 9, Yes], [2, 1, 6, 6, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0007","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"512925fbc056fbe5b30cf3d887f652f549ee2b34dd00793eb3bcf6714be2283f","family_id":"qf-mod47-contract-metric-qualification","seed":202611161007,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 7, 21], [7, 0, 14], [21, 14, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7/2, Yes], [0, 2, 7, 21/2, Yes], [1, 0, 0, 7/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 7, 7, Yes], [2, 0, 7, 21/2, Yes], [2, 1, 7, 7, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7/2, Yes], [0, 2, 7, 21/2, Yes], [1, 0, 0, 7/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 7, 7, Yes], [2, 0, 7, 21/2, Yes], [2, 1, 7, 7, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0008","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cffd5239013b6d6534e8b1da1db3890506dd466638bbc9b451bcd3faef95ee9b","family_id":"qf-mod47-contract-metric-qualification","seed":202611161008,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 4, Yes], [0, 2, 8, 12, Yes], [1, 0, 0, 4, Yes], [1, 1, 0, 0, Yes], [1, 2, 8, 8, Yes], [2, 0, 8, 12, Yes], [2, 1, 8, 8, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 4, Yes], [0, 2, 8, 12, Yes], [1, 0, 0, 4, Yes], [1, 1, 0, 0, Yes], [1, 2, 8, 8, Yes], [2, 0, 8, 12, Yes], [2, 1, 8, 8, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0009","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31ed0f2d0bb24db4951e4045284e4f364c2465df1e3330674ac9f36b7469d4bb","family_id":"qf-mod47-contract-metric-qualification","seed":202611161009,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 9, 27], [9, 0, 18], [27, 18, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9/2, Yes], [0, 2, 9, 27/2, Yes], [1, 0, 0, 9/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 9, 9, Yes], [2, 0, 9, 27/2, Yes], [2, 1, 9, 9, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9/2, Yes], [0, 2, 9, 27/2, Yes], [1, 0, 0, 9/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 9, 9, Yes], [2, 0, 9, 27/2, Yes], [2, 1, 9, 9, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0010","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c1b75e3615d48eef44a768741b203bdb20ee326cf529643db041b5acb363024","family_id":"qf-mod47-contract-metric-qualification","seed":202611161010,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5, Yes], [0, 2, 10, 15, Yes], [1, 0, 0, 5, Yes], [1, 1, 0, 0, Yes], [1, 2, 10, 10, Yes], [2, 0, 10, 15, Yes], [2, 1, 10, 10, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5, Yes], [0, 2, 10, 15, Yes], [1, 0, 0, 5, Yes], [1, 1, 0, 0, Yes], [1, 2, 10, 10, Yes], [2, 0, 10, 15, Yes], [2, 1, 10, 10, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0011","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adec6a4b891d51b3154a24f4ce57378f3be2bb53387e000f189cf2a864b36997","family_id":"qf-mod47-contract-metric-qualification","seed":202611161011,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11/2, Yes], [0, 2, 11, 33/2, Yes], [1, 0, 0, 11/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 11, 11, Yes], [2, 0, 11, 33/2, Yes], [2, 1, 11, 11, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11/2, Yes], [0, 2, 11, 33/2, Yes], [1, 0, 0, 11/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 11, 11, Yes], [2, 0, 11, 33/2, Yes], [2, 1, 11, 11, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0012","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"105e1aa5d88c3e1a89d4da59f917a583ee0a27c89f26dc908b133344c543452f","family_id":"qf-mod47-contract-metric-qualification","seed":202611161012,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 6, Yes], [0, 2, 12, 18, Yes], [1, 0, 0, 6, Yes], [1, 1, 0, 0, Yes], [1, 2, 12, 12, Yes], [2, 0, 12, 18, Yes], [2, 1, 12, 12, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 6, Yes], [0, 2, 12, 18, Yes], [1, 0, 0, 6, Yes], [1, 1, 0, 0, Yes], [1, 2, 12, 12, Yes], [2, 0, 12, 18, Yes], [2, 1, 12, 12, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0013","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a578953689e2a7db67ed5792ec31943368ba7fe7a722e9b44c9fde4a60806f10","family_id":"qf-mod47-contract-metric-qualification","seed":202611161013,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 13/2, Yes], [0, 2, 13, 39/2, Yes], [1, 0, 0, 13/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 13, 13, Yes], [2, 0, 13, 39/2, Yes], [2, 1, 13, 13, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 13/2, Yes], [0, 2, 13, 39/2, Yes], [1, 0, 0, 13/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 13, 13, Yes], [2, 0, 13, 39/2, Yes], [2, 1, 13, 13, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0014","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccad037879ca9c34d6125b07360665746d4d13d04b5b4a0b07c2063c594c4e1c","family_id":"qf-mod47-contract-metric-qualification","seed":202611161014,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7, Yes], [0, 2, 14, 21, Yes], [1, 0, 0, 7, Yes], [1, 1, 0, 0, Yes], [1, 2, 14, 14, Yes], [2, 0, 14, 21, Yes], [2, 1, 14, 14, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7, Yes], [0, 2, 14, 21, Yes], [1, 0, 0, 7, Yes], [1, 1, 0, 0, Yes], [1, 2, 14, 14, Yes], [2, 0, 14, 21, Yes], [2, 1, 14, 14, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0015","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e89a409ffae86d80ea642b6331ae7704c3eafd40c2a513147f4a7abe280b80","family_id":"qf-mod47-contract-metric-qualification","seed":202611161015,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 15/2, Yes], [0, 2, 15, 45/2, Yes], [1, 0, 0, 15/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 15, 15, Yes], [2, 0, 15, 45/2, Yes], [2, 1, 15, 15, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 15/2, Yes], [0, 2, 15, 45/2, Yes], [1, 0, 0, 15/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 15, 15, Yes], [2, 0, 15, 45/2, Yes], [2, 1, 15, 15, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0016","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e202d51a5f6535ad30fd3a8c1da54747f82b162182b1ccadf634b3287a065df","family_id":"qf-mod47-contract-metric-qualification","seed":202611161016,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 8, Yes], [0, 2, 16, 24, Yes], [1, 0, 0, 8, Yes], [1, 1, 0, 0, Yes], [1, 2, 16, 16, Yes], [2, 0, 16, 24, Yes], [2, 1, 16, 16, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 8, Yes], [0, 2, 16, 24, Yes], [1, 0, 0, 8, Yes], [1, 1, 0, 0, Yes], [1, 2, 16, 16, Yes], [2, 0, 16, 24, Yes], [2, 1, 16, 16, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0017","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bc799dcd80d2e3ad3376485cb2a7a4a6dc7466bcecf4fc341dbd1bb03f7bbef","family_id":"qf-mod47-contract-metric-qualification","seed":202611161017,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 17/2, Yes], [0, 2, 17, 51/2, Yes], [1, 0, 0, 17/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 17, 17, Yes], [2, 0, 17, 51/2, Yes], [2, 1, 17, 17, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 17/2, Yes], [0, 2, 17, 51/2, Yes], [1, 0, 0, 17/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 17, 17, Yes], [2, 0, 17, 51/2, Yes], [2, 1, 17, 17, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0018","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6107e51e4b57a0d78937c288fe11c18d16611ed204db0aa71240c67084f517d7","family_id":"qf-mod47-contract-metric-qualification","seed":202611161018,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9, Yes], [0, 2, 18, 27, Yes], [1, 0, 0, 9, Yes], [1, 1, 0, 0, Yes], [1, 2, 18, 18, Yes], [2, 0, 18, 27, Yes], [2, 1, 18, 18, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9, Yes], [0, 2, 18, 27, Yes], [1, 0, 0, 9, Yes], [1, 1, 0, 0, Yes], [1, 2, 18, 18, Yes], [2, 0, 18, 27, Yes], [2, 1, 18, 18, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0019","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0474abc9a10c66a2cfd6de0929b4e806ec67596ef263cf1bc56d34a82d088e43","family_id":"qf-mod47-contract-metric-qualification","seed":202611161019,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 19/2, Yes], [0, 2, 19, 57/2, Yes], [1, 0, 0, 19/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 19, 19, Yes], [2, 0, 19, 57/2, Yes], [2, 1, 19, 19, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 19/2, Yes], [0, 2, 19, 57/2, Yes], [1, 0, 0, 19/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 19, 19, Yes], [2, 0, 19, 57/2, Yes], [2, 1, 19, 19, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0020","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6e005b689ad29ac0a2e56d903adc49856bbacb08b30840fcddcbe1d38aac404","family_id":"qf-mod47-contract-metric-qualification","seed":202611161020,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 10, Yes], [0, 2, 20, 30, Yes], [1, 0, 0, 10, Yes], [1, 1, 0, 0, Yes], [1, 2, 20, 20, Yes], [2, 0, 20, 30, Yes], [2, 1, 20, 20, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 10, Yes], [0, 2, 20, 30, Yes], [1, 0, 0, 10, Yes], [1, 1, 0, 0, Yes], [1, 2, 20, 20, Yes], [2, 0, 20, 30, Yes], [2, 1, 20, 20, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0021","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65feaf42e10f44c2c4cf6127624e28b7fbc919ded3bdc7afaa9b9270d544449d","family_id":"qf-mod47-contract-metric-qualification","seed":202611161021,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 21/2, Yes], [0, 2, 21, 63/2, Yes], [1, 0, 0, 21/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 21, 21, Yes], [2, 0, 21, 63/2, Yes], [2, 1, 21, 21, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 21/2, Yes], [0, 2, 21, 63/2, Yes], [1, 0, 0, 21/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 21, 21, Yes], [2, 0, 21, 63/2, Yes], [2, 1, 21, 21, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0022","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba8b676dc483c10131aeceeb019fc730fd0d03e60d5d4124c1e83fca93ae86f","family_id":"qf-mod47-contract-metric-qualification","seed":202611161022,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11, Yes], [0, 2, 22, 33, Yes], [1, 0, 0, 11, Yes], [1, 1, 0, 0, Yes], [1, 2, 22, 22, Yes], [2, 0, 22, 33, Yes], [2, 1, 22, 22, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11, Yes], [0, 2, 22, 33, Yes], [1, 0, 0, 11, Yes], [1, 1, 0, 0, Yes], [1, 2, 22, 22, Yes], [2, 0, 22, 33, Yes], [2, 1, 22, 22, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0023","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2ce259f51cd73c9eb021872edeb38ad3dafa17f5d4f5423640d73de6d1ba5c3","family_id":"qf-mod47-contract-metric-qualification","seed":202611161023,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 23/2, Yes], [0, 2, 23, 69/2, Yes], [1, 0, 0, 23/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 23, 23, Yes], [2, 0, 23, 69/2, Yes], [2, 1, 23, 23, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 23/2, Yes], [0, 2, 23, 69/2, Yes], [1, 0, 0, 23/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 23, 23, Yes], [2, 0, 23, 69/2, Yes], [2, 1, 23, 23, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0024","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e36a56274f441737deec02f0cf3adc5f739fc37749b9722ca763bb598d34dfaa","family_id":"qf-mod47-contract-metric-qualification","seed":202611161024,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 12, Yes], [0, 2, 24, 36, Yes], [1, 0, 0, 12, Yes], [1, 1, 0, 0, Yes], [1, 2, 24, 24, Yes], [2, 0, 24, 36, Yes], [2, 1, 24, 24, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 12, Yes], [0, 2, 24, 36, Yes], [1, 0, 0, 12, Yes], [1, 1, 0, 0, Yes], [1, 2, 24, 24, Yes], [2, 0, 24, 36, Yes], [2, 1, 24, 24, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0025","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27bd9d1a4f2a959e919a035883053c199e9ae1a2b6c86addb48f0d39ac38bd0f","family_id":"qf-mod47-contract-metric-qualification","seed":202611161025,"semantic_template":"qf-mod47-contract-metric-qualification/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2]}"},"fields":{"title":"A Global Contraction Needs Every Metric and Mapping Qualification","stem":"Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).","answer":{"kind":"expression","value":"(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 25/2, Yes], [0, 2, 25, 75/2, Yes], [1, 0, 0, 25/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 25, 25, Yes], [2, 0, 25, 75/2, Yes], [2, 1, 25, 25, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)","explanation":"Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 25/2, Yes], [0, 2, 25, 75/2, Yes], [1, 0, 0, 25/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 25, 25, Yes], [2, 0, 25, 75/2, Yes], [2, 1, 25, 25, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)."},"skills":["A Global Contraction Needs Every Metric and Mapping Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0026","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea5527f28bf6daae1253edccb5135b48030cb494e5855a28830568f935d45a9e","family_id":"qf-mod47-contract-picard-orbit","seed":202611161026,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0027","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93f3df52ee1c3d6f1c2366fe629ccf0e5f018d877950f493bd3a653fb3af9a8c","family_id":"qf-mod47-contract-picard-orbit","seed":202611161027,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0028","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ea5bb42137e98ddf0f2625bad04b151739e7ba7ebbe8bbaf69699db0647712","family_id":"qf-mod47-contract-picard-orbit","seed":202611161028,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0029","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72ff62862ac3394298392ae8ed7bf86a2d3e83c75f1085d8099ec7fd684ab37a","family_id":"qf-mod47-contract-picard-orbit","seed":202611161029,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0030","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b2f0bab35569f2f35108d3996fd326d2de7875f5ab480ce3b2abfe9b8817618","family_id":"qf-mod47-contract-picard-orbit","seed":202611161030,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0031","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3c2b771150c51bdd8ba7f27dea5a2c4fd9d2d24ae2bf5c44c8b36d7ec45fa32","family_id":"qf-mod47-contract-picard-orbit","seed":202611161031,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0032","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7464e7031ef7602d41c4ddf8acff72f603030bc1bf667575ff8486b07d656078","family_id":"qf-mod47-contract-picard-orbit","seed":202611161032,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 7, 21], [7, 0, 14], [21, 14, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0033","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"063a4a0588529bffb948684e7beea33aec305ae49d3482529a6fab50f6b7eeaf","family_id":"qf-mod47-contract-picard-orbit","seed":202611161033,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0034","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d214542c8b3f7a7bd5492af5073d09c1baad9e464467696245f83841cbfe8d52","family_id":"qf-mod47-contract-picard-orbit","seed":202611161034,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 9, 27], [9, 0, 18], [27, 18, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0035","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"226ae5ad25bb9eb16d4b2a51b53f22b2bdb41f0550e1394d26ae1dc004ddd495","family_id":"qf-mod47-contract-picard-orbit","seed":202611161035,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0036","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cab36c3c97c7121906279a66b986dd4fd79db083f9f21a92533fba8b9dd8f27a","family_id":"qf-mod47-contract-picard-orbit","seed":202611161036,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0037","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"106151cd55b84b45dec1cabcf9ed71f6d40cd0069bedc01143a091b2392aa86f","family_id":"qf-mod47-contract-picard-orbit","seed":202611161037,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0038","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3c2a98387b9251a4373097f46d331349a07f5dabe6ea373fc3acb28bed9e8a4","family_id":"qf-mod47-contract-picard-orbit","seed":202611161038,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0039","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8652b78898846b77c147facd441b6e9c4144b3d6631083369fc99a0edceeb29c","family_id":"qf-mod47-contract-picard-orbit","seed":202611161039,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0040","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18b1c9d171b9fbb4859bc9f859416862fb855b6e2947d1937d30836fc09fa3ea","family_id":"qf-mod47-contract-picard-orbit","seed":202611161040,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0041","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70bea53b685570fb6c89215d7a7a7e748064be5b9f2142f47badaf630a9364b5","family_id":"qf-mod47-contract-picard-orbit","seed":202611161041,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0042","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2884b2ef765093b8e4ad170495079996b320700d8f86796dcbbc7316456d0b44","family_id":"qf-mod47-contract-picard-orbit","seed":202611161042,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0043","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"660a8a8c2591a2b4f4848318c90a669a80024fd409655b57c312092eb15cf8a3","family_id":"qf-mod47-contract-picard-orbit","seed":202611161043,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0044","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f8e5caf389136570cc612efbbf0852410f35b05ca9652fa209d45383d0e6505","family_id":"qf-mod47-contract-picard-orbit","seed":202611161044,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0045","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"795ccd468a1ea9ee22316792140550e1596eda0f6a1e2e4245f1cee5d6fbdd5c","family_id":"qf-mod47-contract-picard-orbit","seed":202611161045,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0046","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86433e379a3cbc4fad7d3cfe8392813b3109fe29d1f60539fb883c89120616fa","family_id":"qf-mod47-contract-picard-orbit","seed":202611161046,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0047","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2f865c1fca65cc2a00387d0c25056291a9f1aa2c838b3d59ca90ef80682d262","family_id":"qf-mod47-contract-picard-orbit","seed":202611161047,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0048","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e20fac565b283319d14252096421bb3bf5ebfc0072f84e9dc121f176d1ffd76","family_id":"qf-mod47-contract-picard-orbit","seed":202611161048,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0049","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f2c0a8c39495d2df11a706f755d29129930348b012005159576e96b4769bb9","family_id":"qf-mod47-contract-picard-orbit","seed":202611161049,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0050","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"288a221fea89767218f37eeab2c2e50ae31bc4c2f73572a158142026d673e322","family_id":"qf-mod47-contract-picard-orbit","seed":202611161050,"semantic_template":"qf-mod47-contract-picard-orbit/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"Generate the Complete Picard Orbit and Compare with the Unique Fixed Point","stem":"Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).","answer":{"kind":"expression","value":"(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit)","explanation":"Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit)."},"skills":["Generate the Complete Picard Orbit and Compare with the Unique Fixed Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0051","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1cd08bb78ff270d8380551166b7abe358cadcfa775b82102d1682f9f750446","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161051,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[1,1],[3,1]],[[1,1],[0,1],[2,1]],[[3,1],[2,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0052","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb1d354fa198ec86c2139dac2842e94b843bd81685e7adb701de958761961cb2","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161052,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[2,1],[6,1]],[[2,1],[0,1],[4,1]],[[6,1],[4,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0053","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e60ab64b5796ac1eee2469b8a17c7b551901639a7404bf42b8c19b772024f6a","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161053,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[3,1],[9,1]],[[3,1],[0,1],[6,1]],[[9,1],[6,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 3/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 3/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0054","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce4848700c3cb2a0ae6fc9bfc06db9599a56695365d8734d94df9586b8f8f465","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161054,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[4,1],[12,1]],[[4,1],[0,1],[8,1]],[[12,1],[8,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 1/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 1/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0055","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d45393f0663475f771115b3e3bb66e23f9d415b89b2d2f4471d2cacc5af214f3","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161055,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[5,1],[15,1]],[[5,1],[0,1],[10,1]],[[15,1],[10,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0056","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e2cb08d657418075cc5e34fe712f74ddd741d067d4ec5c008d2a5319366496","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161056,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[6,1],[18,1]],[[6,1],[0,1],[12,1]],[[18,1],[12,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0057","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b5bdd0526b0d5d43a33577c892612c867c29e7a23372bc99320e654bad880f9","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161057,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[7,1],[21,1]],[[7,1],[0,1],[14,1]],[[21,1],[14,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 7, 21], [7, 0, 14], [21, 14, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 7/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 7/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0058","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"176219ca5950121da4a4d606b8f941f697f4e8277c67f4025171fc878362fd12","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161058,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[8,1],[24,1]],[[8,1],[0,1],[16,1]],[[24,1],[16,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0059","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4a66799aadf41bb07bad47aeb3b9b864ee1cff25e1db14e159b512b68b87634","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161059,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[9,1],[27,1]],[[9,1],[0,1],[18,1]],[[27,1],[18,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 9, 27], [9, 0, 18], [27, 18, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 9/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 9/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0060","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8af47539ee1b36f93cda57a3cfca21155c75a7d304ff8259ad0c835892eca6de","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161060,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0061","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a42d4872bbe03ee1ffe9634c20d7af0a2e7fba749eefb82273237190a334b13a","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161061,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[11,1],[33,1]],[[11,1],[0,1],[22,1]],[[33,1],[22,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0062","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ff13a360a1661ca34d0303baba244be93ac2a9d9d810655e0233d992239562","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161062,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[12,1],[36,1]],[[12,1],[0,1],[24,1]],[[36,1],[24,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0063","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d17ddec21bd4c99e9dd294196dfbd639436d57c3dad7946320f8f904800ffb9","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161063,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[13,1],[39,1]],[[13,1],[0,1],[26,1]],[[39,1],[26,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 13/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 13/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0064","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cac933c605d108da52f5653df57d8ae5b010a26c2c9160ae3383fcff2f87664d","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161064,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[14,1],[42,1]],[[14,1],[0,1],[28,1]],[[42,1],[28,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 7/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 7/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0065","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfb86a554f829b9b98429dd2d102103757359a92b9957ce0c6e5ee3b3115b784","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161065,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[15,1],[45,1]],[[15,1],[0,1],[30,1]],[[45,1],[30,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0066","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5159c466fcddb4b3f29202def70f4087e2944d1f1c2e5b6ac04ff0f6ba84cf3","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161066,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[16,1],[48,1]],[[16,1],[0,1],[32,1]],[[48,1],[32,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0067","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c22b618468fc0c268e600da5887e61fbf246b586439e4d2a754c1e9a2d74347","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161067,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[17,1],[51,1]],[[17,1],[0,1],[34,1]],[[51,1],[34,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 17/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 17/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0068","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e1fc5328d8e50aab6780eb5ac26f3a36ea6ce982c50c333ed8ad09e35d81365","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161068,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[18,1],[54,1]],[[18,1],[0,1],[36,1]],[[54,1],[36,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 9/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 9/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0069","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef9e1918e10daf200d2693c2a8b77d06410ab323740274fb4122ab63e2f2f12d","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161069,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[19,1],[57,1]],[[19,1],[0,1],[38,1]],[[57,1],[38,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 19/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 19/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0070","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43ae4d5931b831807bedfebd46116f6a48764cbc745204d7ac0671d10e9acfd2","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161070,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[20,1],[60,1]],[[20,1],[0,1],[40,1]],[[60,1],[40,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0071","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa8adcb013843a8b4aa42bec1386daac1b186fbdb8d0f0e980e9b079c1855f00","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161071,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[21,1],[63,1]],[[21,1],[0,1],[42,1]],[[63,1],[42,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":2}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0], 0, 0, 21, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 21, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0072","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe078f6739e539a5d6a0883d9b117cdf96289eeb865237f26af4b62ece83bc4d","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161072,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[22,1],[66,1]],[[22,1],[0,1],[44,1]],[[66,1],[44,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":3}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0073","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f85d9a61af832c4514804157833c8456691844620f6fd58e44cf2bc59c6b316","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161073,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[23,1],[69,1]],[[23,1],[0,1],[46,1]],[[69,1],[46,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":4}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0], 0, 0, 23/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 23/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0074","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57e0dca1d29862dc53fa14bab098579c302f0ac52e55ef0ab3617c0a450a5e9e","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161074,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[24,1],[72,1]],[[24,1],[0,1],[48,1]],[[72,1],[48,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":5}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0], 0, 0, 3, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 3, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0075","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d4d8da97bf077df5821ffe99941c7a2b91af5b3870d116c647174be52c7d619","family_id":"qf-mod47-contract-fixed-point-bound","seed":202611161075,"semantic_template":"qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[25,1],[75,1]],[[25,1],[0,1],[50,1]],[[75,1],[50,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}"},"fields":{"title":"The Geometric Tail Bound Controls the Exact Picard Error","stem":"Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)","explanation":"Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)."},"skills":["The Geometric Tail Bound Controls the Exact Picard Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0076","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ae56ab2869aa261c1847c3ee7e58ac17bb4ea9c1b3b04a1f92788ce6c52caa","family_id":"qf-mod47-contract-boundary","seed":202611161076,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[1,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=1. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0077","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06e05ff075b518aa42ea4d83e99ef204b7035b551314423bb578102287d58aff","family_id":"qf-mod47-contract-boundary","seed":202611161077,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[2,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=2. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 2, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 2, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0078","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b64360e4018cb2760319d667239761a49e46b5a7c80b1aa8bcfd44731610ad","family_id":"qf-mod47-contract-boundary","seed":202611161078,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[3,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=3. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0079","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ede96171b69ab81673ba19b28f51e3bb88431b2034ce919f47fe1ecb6f32ec3","family_id":"qf-mod47-contract-boundary","seed":202611161079,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[4,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=4. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0080","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14bd5d3cb50f9a36c9003e6b007d8d84b0da3713c5bb585e3b1786a0b3ac508","family_id":"qf-mod47-contract-boundary","seed":202611161080,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[5,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=5. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0081","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd162231b0c204b377dd234cc5cb559c070252e9552ec3065f01076b5575d7b0","family_id":"qf-mod47-contract-boundary","seed":202611161081,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[6,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=6. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 6, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 6, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0082","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4522d593504357b72866411866b420343cb6c21f036b6edfcf8521740d95c64","family_id":"qf-mod47-contract-boundary","seed":202611161082,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[7,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=7. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0083","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92630af11c3f3e6573ad7b220ea4a0c97479cb4bd9aca1dfe53366a1f2699e68","family_id":"qf-mod47-contract-boundary","seed":202611161083,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[8,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=8. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0084","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da5cf987869b313f7e231c4ad71fae7158eec5a14c97f1e437c144d885b80029","family_id":"qf-mod47-contract-boundary","seed":202611161084,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[9,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=9. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0085","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0977eac0ff342965281902d788d5a36876b297c7ca9c231034d42f84f1d349a3","family_id":"qf-mod47-contract-boundary","seed":202611161085,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[10,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=10. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 10, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 10, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0086","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a347d1ee99f5f9f575d4470d21dd06ba8c3fb547b5059a5a567addf45188aba7","family_id":"qf-mod47-contract-boundary","seed":202611161086,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[11,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=11. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0087","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"177fadd763d7fc1a2b11d99c47d8edc698facd951569ecab4983c9a8ebe8f26a","family_id":"qf-mod47-contract-boundary","seed":202611161087,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[12,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=12. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0088","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5426573dd0653ce3c36ec52e79f24caf2bf80e7bac1126cbfca9ec83d342c649","family_id":"qf-mod47-contract-boundary","seed":202611161088,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[13,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=13. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0089","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"615c477d1bc99a80239a45b84f34e8f54a75ab7455a7277ebbac183a54af4c7a","family_id":"qf-mod47-contract-boundary","seed":202611161089,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[14,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=14. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 14, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 14, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0090","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc7dc1de2c11df5b42653b40407d2828ca1c05a3aa98a09ee6aaa79517b131b8","family_id":"qf-mod47-contract-boundary","seed":202611161090,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[15,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=15. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0091","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9925cf5223aadf1dbf91dc24aabf0cf0accc3f32a839628e384e0ad0ea52c3","family_id":"qf-mod47-contract-boundary","seed":202611161091,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[16,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=16. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0092","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1408db42f69d02fbfb67ce9356a2ebf6fe963769a923e47e3065a740fe03b00","family_id":"qf-mod47-contract-boundary","seed":202611161092,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[17,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=17. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0093","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5e1af6637331e36e0c44947d893007bc39b32500884a5d6d90e2576d608905f","family_id":"qf-mod47-contract-boundary","seed":202611161093,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[18,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=18. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 18, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 18, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0094","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48a3f29c5911ecf1f085a8b9785c6ebb65b054de22fe4c8c3288520f654b3b97","family_id":"qf-mod47-contract-boundary","seed":202611161094,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[19,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=19. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0095","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb3e25ec8a93860ae1bd23b16135e0437347d0b8d58f8a03142b1a138c0efccc","family_id":"qf-mod47-contract-boundary","seed":202611161095,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[20,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=20. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0096","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed1a28b9d48d338395b20ba180603eefb97d2c17e0de4c15ed44d321dc68bdbe","family_id":"qf-mod47-contract-boundary","seed":202611161096,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[21,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=21. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0097","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba3061afcc38ab144950753dd6d1332d548d0ddefccc2db7972d5bdfa529df4a","family_id":"qf-mod47-contract-boundary","seed":202611161097,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"non_self_map\",\"scale\":[22,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=non_self_map and positive exact scale=22. Give (mode, scale, mapping_description, theorem_application_claim).","answer":{"kind":"expression","value":"(non_self_map, 22, T_maps_one_point_outside_X, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 22, T_maps_one_point_outside_X, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0098","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b138e699eb4f7449450c45a989005674a986dce988ccf49eab02e86ea853f62","family_id":"qf-mod47-contract-boundary","seed":202611161098,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[23,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=noncomplete_space and positive exact scale=23. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).","answer":{"kind":"expression","value":"(noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0099","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1669a3f34b27d3a18fa20f536ccab629b008872942e3b24e749eb86920bb52d","family_id":"qf-mod47-contract-boundary","seed":202611161099,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"sampled_pairs\",\"scale\":[24,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=sampled_pairs and positive exact scale=24. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).","answer":{"kind":"expression","value":"(sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r01-contraction-mapping-theorem-0100","batch_id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6036732d61294cb41e00251124e805ff499c73d2b71ab8094affd9ddd50ba34b","family_id":"qf-mod47-contract-boundary","seed":202611161100,"semantic_template":"qf-mod47-contract-boundary/{\"mode\":\"q_equal_one\",\"scale\":[25,1]}"},"fields":{"title":"Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem","stem":"Use boundary mode=q_equal_one and positive exact scale=25. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).","answer":{"kind":"expression","value":"(q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, No)","explanation":"For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, No)."},"skills":["Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}