{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r02-0001","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a32bea0092710c0a64f66a9d8f15bc6016f4fa7d423b3b785d9141479658613","family_id":"qf-mod50-trap-partition-qualification","seed":202611192001,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[3,1]],\"values\":[[0,1],[1,1],[3,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 3] and aligned endpoint values=[0, 1, 3]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 3], [0, 1, 3], 2, 3, [1, 2], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], 2, 3, [1, 2], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0002","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b25da9b94317a377cdd278130b9962c951d43d405167274737f9d7dccec6039","family_id":"qf-mod50-trap-partition-qualification","seed":202611192002,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[4,1]],\"values\":[[1,1],[2,1],[4,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 4] and aligned endpoint values=[1, 2, 4]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 4], [1, 2, 4], 2, 3, [1, 3], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 4], [1, 2, 4], 2, 3, [1, 3], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0003","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8942e47fb8f056794482ed1b96bdb6e6362d57ce8de1a971f08eb2ff6ac9fd4b","family_id":"qf-mod50-trap-partition-qualification","seed":202611192003,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[5,1]],\"values\":[[2,1],[3,1],[5,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 5] and aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 5], [2, 3, 5], 2, 3, [1, 4], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 5], [2, 3, 5], 2, 3, [1, 4], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0004","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e145311151d65e4bc9cf06cf25c3c0b6dfaff68a57ffe494f9fa77d138d68cf","family_id":"qf-mod50-trap-partition-qualification","seed":202611192004,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[6,1]],\"values\":[[3,1],[4,1],[6,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 6] and aligned endpoint values=[3, 4, 6]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 6], [3, 4, 6], 2, 3, [1, 5], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 6], [3, 4, 6], 2, 3, [1, 5], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0005","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"840aa3bae9d5fc0923eacb148729ba092b6c112cac94cb89d7c3edd01bcfdcb9","family_id":"qf-mod50-trap-partition-qualification","seed":202611192005,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[7,1]],\"values\":[[4,1],[5,1],[7,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 7] and aligned endpoint values=[4, 5, 7]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 7], [4, 5, 7], 2, 3, [1, 6], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 7], [4, 5, 7], 2, 3, [1, 6], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0006","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5725c27cd8685388f9980a92c2c5a8025c791e1a64fd8b1633d9b6ced6a8c23a","family_id":"qf-mod50-trap-partition-qualification","seed":202611192006,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[8,1]],\"values\":[[5,1],[6,1],[8,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 8] and aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 8], [5, 6, 8], 2, 3, [1, 7], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 8], [5, 6, 8], 2, 3, [1, 7], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0007","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8aef91043a6e05d745a73a00fbcbca4ca1859a6a48f0002bd7e4b39c7ba462c","family_id":"qf-mod50-trap-partition-qualification","seed":202611192007,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[9,1]],\"values\":[[6,1],[7,1],[9,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 9] and aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 9], [6, 7, 9], 2, 3, [1, 8], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 9], [6, 7, 9], 2, 3, [1, 8], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0008","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a5cc5b0b003a09dc21e3ce728ae10fac412e9ce3562ccac9e1f08871c8f6d3","family_id":"qf-mod50-trap-partition-qualification","seed":202611192008,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[10,1]],\"values\":[[7,1],[8,1],[10,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 10] and aligned endpoint values=[7, 8, 10]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 10], [7, 8, 10], 2, 3, [1, 9], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 10], [7, 8, 10], 2, 3, [1, 9], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0009","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31b83593d677ddfd6a427d69abf9e5e1baea6c4174b0b18dba30d86f19708a62","family_id":"qf-mod50-trap-partition-qualification","seed":202611192009,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[11,1]],\"values\":[[8,1],[9,1],[11,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 11] and aligned endpoint values=[8, 9, 11]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 11], [8, 9, 11], 2, 3, [1, 10], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 11], [8, 9, 11], 2, 3, [1, 10], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0010","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c702cc4d68cd8388e1ac24acfb2f2666e807f92ce37ef99eb38e8b4763cde9d6","family_id":"qf-mod50-trap-partition-qualification","seed":202611192010,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[12,1]],\"values\":[[9,1],[10,1],[12,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 12] and aligned endpoint values=[9, 10, 12]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 12], [9, 10, 12], 2, 3, [1, 11], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 12], [9, 10, 12], 2, 3, [1, 11], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0011","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7702510cab8d6541672ac7e731ce2744c1a8827b9f7d991b500925283257c1d3","family_id":"qf-mod50-trap-partition-qualification","seed":202611192011,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[13,1]],\"values\":[[10,1],[11,1],[13,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 13] and aligned endpoint values=[10, 11, 13]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 13], [10, 11, 13], 2, 3, [1, 12], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 13], [10, 11, 13], 2, 3, [1, 12], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0012","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"043a63baeb3834ef6b8df2fe6aecd6dc1e595b131322c40f6e8b80eb3aa3a1b7","family_id":"qf-mod50-trap-partition-qualification","seed":202611192012,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[14,1]],\"values\":[[11,1],[12,1],[14,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 14] and aligned endpoint values=[11, 12, 14]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 14], [11, 12, 14], 2, 3, [1, 13], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 14], [11, 12, 14], 2, 3, [1, 13], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0013","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"803b606d41946a3c5691b2a9741b34ea4c717c4d86e74c83efe71b734b382281","family_id":"qf-mod50-trap-partition-qualification","seed":202611192013,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[15,1]],\"values\":[[12,1],[13,1],[15,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 15] and aligned endpoint values=[12, 13, 15]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 15], [12, 13, 15], 2, 3, [1, 14], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 15], [12, 13, 15], 2, 3, [1, 14], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0014","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdecdb7dacb35f2a50d0d270b3c409c1756fc5695a2ef13032a090667696d7ef","family_id":"qf-mod50-trap-partition-qualification","seed":202611192014,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[16,1]],\"values\":[[13,1],[14,1],[16,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 16] and aligned endpoint values=[13, 14, 16]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 16], [13, 14, 16], 2, 3, [1, 15], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 16], [13, 14, 16], 2, 3, [1, 15], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0015","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"759d415fa098adc56fa93b0592d2017094ba91082c7767bc54ec12c5a431c0e2","family_id":"qf-mod50-trap-partition-qualification","seed":202611192015,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[17,1]],\"values\":[[14,1],[15,1],[17,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 17] and aligned endpoint values=[14, 15, 17]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 17], [14, 15, 17], 2, 3, [1, 16], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 17], [14, 15, 17], 2, 3, [1, 16], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0016","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ea9183c5b0a97f47aabd2944630ce2aeb8f6aa0f7be748d42c568c678800e3","family_id":"qf-mod50-trap-partition-qualification","seed":202611192016,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[18,1]],\"values\":[[15,1],[16,1],[18,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 18] and aligned endpoint values=[15, 16, 18]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 18], [15, 16, 18], 2, 3, [1, 17], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 18], [15, 16, 18], 2, 3, [1, 17], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0017","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0d824605b3a463e5ec8d0a402eeaece9802f729d06bb1427fe6548b56c1730","family_id":"qf-mod50-trap-partition-qualification","seed":202611192017,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[19,1]],\"values\":[[16,1],[17,1],[19,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 19] and aligned endpoint values=[16, 17, 19]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 19], [16, 17, 19], 2, 3, [1, 18], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 19], [16, 17, 19], 2, 3, [1, 18], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0018","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60ec1260b7819364d63598b4615238940f04e3ca1912a7cc25a2558259b980a3","family_id":"qf-mod50-trap-partition-qualification","seed":202611192018,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[20,1]],\"values\":[[17,1],[18,1],[20,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 20] and aligned endpoint values=[17, 18, 20]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 20], [17, 18, 20], 2, 3, [1, 19], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 20], [17, 18, 20], 2, 3, [1, 19], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0019","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2bf0f0466a182cb646142c4a7023c33772f7ae7801e6a345e9ca75546797f66","family_id":"qf-mod50-trap-partition-qualification","seed":202611192019,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[21,1]],\"values\":[[18,1],[19,1],[21,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 21] and aligned endpoint values=[18, 19, 21]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 21], [18, 19, 21], 2, 3, [1, 20], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 21], [18, 19, 21], 2, 3, [1, 20], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0020","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc85aa52122e96dc35634f12449804efc2ae2df2549d508950f1cd87ee7dc8b9","family_id":"qf-mod50-trap-partition-qualification","seed":202611192020,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[22,1]],\"values\":[[19,1],[20,1],[22,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 22] and aligned endpoint values=[19, 20, 22]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 22], [19, 20, 22], 2, 3, [1, 21], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 22], [19, 20, 22], 2, 3, [1, 21], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0021","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f58974f2e0e18ccd50cd31567fcb5caa4adb6d083b28641c273e87b6f97a1d66","family_id":"qf-mod50-trap-partition-qualification","seed":202611192021,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[23,1]],\"values\":[[20,1],[21,1],[23,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 23] and aligned endpoint values=[20, 21, 23]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 23], [20, 21, 23], 2, 3, [1, 22], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 23], [20, 21, 23], 2, 3, [1, 22], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0022","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a837d8b3b6fe317a4c554da530594a196e41c0fdfef75425ac025ad7eace5d1f","family_id":"qf-mod50-trap-partition-qualification","seed":202611192022,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[24,1]],\"values\":[[21,1],[22,1],[24,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 24] and aligned endpoint values=[21, 22, 24]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 24], [21, 22, 24], 2, 3, [1, 23], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 24], [21, 22, 24], 2, 3, [1, 23], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0023","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83237e574e88924d4eebbd6a4f7b9e95d0cfe9a75661b639f05b7c20abd8713a","family_id":"qf-mod50-trap-partition-qualification","seed":202611192023,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[25,1]],\"values\":[[22,1],[23,1],[25,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 25] and aligned endpoint values=[22, 23, 25]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 25], [22, 23, 25], 2, 3, [1, 24], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 25], [22, 23, 25], 2, 3, [1, 24], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0024","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d95e7d55f84a01dd6c9e4445f7b222de5478208c2adb7ea00fab6abd5b37174c","family_id":"qf-mod50-trap-partition-qualification","seed":202611192024,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[26,1]],\"values\":[[23,1],[24,1],[26,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 26] and aligned endpoint values=[23, 24, 26]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 26], [23, 24, 26], 2, 3, [1, 25], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 26], [23, 24, 26], 2, 3, [1, 25], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0025","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c3affde9d1a9d076674a6f2d38d6805acfed2eb4a7b1f284a04f3d712309cdd","family_id":"qf-mod50-trap-partition-qualification","seed":202611192025,"semantic_template":"trap-partition-qualification|{\"nodes\":[[0,1],[1,1],[27,1]],\"values\":[[24,1],[25,1],[27,1]]}"},"fields":{"title":"n Subintervals Require n+1 Aligned Strictly Increasing Nodes","stem":"Use ordered nodes=[0, 1, 27] and aligned endpoint values=[24, 25, 27]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).","answer":{"kind":"expression","value":"([0, 1, 27], [24, 25, 27], 2, 3, [1, 26], Yes, n_subintervals_have_n_plus_one_nodes)","explanation":"Check every node/value association and strict node increase. Retain the count of subintervals, count of nodes and every individual width. Do not sort an invalid partition into a valid one or replace a nonuniform partition with one global width. The exact result, retaining the requested order, is ([0, 1, 27], [24, 25, 27], 2, 3, [1, 26], Yes, n_subintervals_have_n_plus_one_nodes)."},"skills":["n Subintervals Require n+1 Aligned Strictly Increasing Nodes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0026","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10127dce17ff9b9e0e566b5ba6690b0df433fcff5401daa7c9d69a04d1a63e93","family_id":"qf-mod50-trap-composite-value","seed":202611192026,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[3,1]],\"values\":[[0,1],[1,1],[3,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 3], aligned endpoint values=[0, 1, 3]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 3], [0, 1, 3], [1/2, 4], 9/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], [1/2, 4], 9/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0027","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91594fb4e62b0b8c4248ad84a98beb1565edaa64eb15af9c841c7b62f26d20da","family_id":"qf-mod50-trap-composite-value","seed":202611192027,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[4,1]],\"values\":[[1,1],[2,1],[4,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 4], aligned endpoint values=[1, 2, 4]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 4], [1, 2, 4], [3/2, 9], 21/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 4], [1, 2, 4], [3/2, 9], 21/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0028","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca504fbd1a46c165cbce3ec7b20c6e8fe3c5417bfc962ef2500f839d7d16105b","family_id":"qf-mod50-trap-composite-value","seed":202611192028,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[5,1]],\"values\":[[2,1],[3,1],[5,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 5], aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 5], [2, 3, 5], [5/2, 16], 37/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 5], [2, 3, 5], [5/2, 16], 37/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0029","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"975a4c9e0b3828574cfa526835e975f7d601a0d094af9bd80c9089777e8a9be1","family_id":"qf-mod50-trap-composite-value","seed":202611192029,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[6,1]],\"values\":[[3,1],[4,1],[6,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 6], aligned endpoint values=[3, 4, 6]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 6], [3, 4, 6], [7/2, 25], 57/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 6], [3, 4, 6], [7/2, 25], 57/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0030","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4789d1d8783902c188c8766d56b2a6788e61e901e4a8ac79b646cf001d18d87","family_id":"qf-mod50-trap-composite-value","seed":202611192030,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[7,1]],\"values\":[[4,1],[5,1],[7,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 7], aligned endpoint values=[4, 5, 7]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 7], [4, 5, 7], [9/2, 36], 81/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 7], [4, 5, 7], [9/2, 36], 81/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0031","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2671f0a835a5378392782379a3902bb693ce51dc03f7f323def5713d717d74c0","family_id":"qf-mod50-trap-composite-value","seed":202611192031,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[8,1]],\"values\":[[5,1],[6,1],[8,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 8], aligned endpoint values=[5, 6, 8]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 8], [5, 6, 8], [11/2, 49], 109/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 8], [5, 6, 8], [11/2, 49], 109/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0032","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30335bd4d948502a9323d04d90c00f06d39ba013666d85ef75b0b6050071e57e","family_id":"qf-mod50-trap-composite-value","seed":202611192032,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[9,1]],\"values\":[[6,1],[7,1],[9,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 9], aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 9], [6, 7, 9], [13/2, 64], 141/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 9], [6, 7, 9], [13/2, 64], 141/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0033","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42c56aad16b248b1cbddc5f8ec03bd48403ad1d2c88f43353af7fc5444bd604c","family_id":"qf-mod50-trap-composite-value","seed":202611192033,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[10,1]],\"values\":[[7,1],[8,1],[10,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 10], aligned endpoint values=[7, 8, 10]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 10], [7, 8, 10], [15/2, 81], 177/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 10], [7, 8, 10], [15/2, 81], 177/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0034","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7ca9bc95454805b5e6f77aa950d6c166bad4d4a5b7aac0b9459bcb9fae67fea","family_id":"qf-mod50-trap-composite-value","seed":202611192034,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[11,1]],\"values\":[[8,1],[9,1],[11,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 11], aligned endpoint values=[8, 9, 11]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 11], [8, 9, 11], [17/2, 100], 217/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 11], [8, 9, 11], [17/2, 100], 217/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0035","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8212f56c1f6702a65f14355927426174a45ff396e782309996ad0918e14b74ce","family_id":"qf-mod50-trap-composite-value","seed":202611192035,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[12,1]],\"values\":[[9,1],[10,1],[12,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 12], aligned endpoint values=[9, 10, 12]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 12], [9, 10, 12], [19/2, 121], 261/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 12], [9, 10, 12], [19/2, 121], 261/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0036","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3e2fce0197b97abb6ad30586c29fe95b8f03b3ac41c6a17e5460e5a2f23756c","family_id":"qf-mod50-trap-composite-value","seed":202611192036,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[13,1]],\"values\":[[10,1],[11,1],[13,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 13], aligned endpoint values=[10, 11, 13]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 13], [10, 11, 13], [21/2, 144], 309/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 13], [10, 11, 13], [21/2, 144], 309/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0037","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d06bd9fa720f705f369eea348a6fab899f677b762909ad94292b9f554f29d5de","family_id":"qf-mod50-trap-composite-value","seed":202611192037,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[14,1]],\"values\":[[11,1],[12,1],[14,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 14], aligned endpoint values=[11, 12, 14]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 14], [11, 12, 14], [23/2, 169], 361/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 14], [11, 12, 14], [23/2, 169], 361/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0038","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da80ba9eac2e77966a7bd94b4d4122013b88915d64b129e6ddae77d023011258","family_id":"qf-mod50-trap-composite-value","seed":202611192038,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[15,1]],\"values\":[[12,1],[13,1],[15,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 15], aligned endpoint values=[12, 13, 15]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 15], [12, 13, 15], [25/2, 196], 417/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 15], [12, 13, 15], [25/2, 196], 417/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0039","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a372153dd9b0ce0dc8668236e5a5d1ea287fa59835b4e1e156c32afef89ba1b","family_id":"qf-mod50-trap-composite-value","seed":202611192039,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[16,1]],\"values\":[[13,1],[14,1],[16,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 16], aligned endpoint values=[13, 14, 16]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 16], [13, 14, 16], [27/2, 225], 477/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 16], [13, 14, 16], [27/2, 225], 477/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0040","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75f49d0356ffa3a68811a9a715a1203b6693962d5b65931bf5d32571ede96a72","family_id":"qf-mod50-trap-composite-value","seed":202611192040,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[17,1]],\"values\":[[14,1],[15,1],[17,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 17], aligned endpoint values=[14, 15, 17]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 17], [14, 15, 17], [29/2, 256], 541/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 17], [14, 15, 17], [29/2, 256], 541/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0041","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca780190ee61be8da8db2b7ea881789c11eeeb8a4cf1edc1d609851f65f4a3b7","family_id":"qf-mod50-trap-composite-value","seed":202611192041,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[18,1]],\"values\":[[15,1],[16,1],[18,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 18], aligned endpoint values=[15, 16, 18]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 18], [15, 16, 18], [31/2, 289], 609/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 18], [15, 16, 18], [31/2, 289], 609/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0042","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5c8ca1613c8657e3d3817c89c4d45de3d9a61a8ac8af38f742a5a78904f2437","family_id":"qf-mod50-trap-composite-value","seed":202611192042,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[19,1]],\"values\":[[16,1],[17,1],[19,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 19], aligned endpoint values=[16, 17, 19]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 19], [16, 17, 19], [33/2, 324], 681/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 19], [16, 17, 19], [33/2, 324], 681/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0043","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e4a90da86642a4d4fff662c4cdc3519f9006c469e0153936305d2e55dae7ee3","family_id":"qf-mod50-trap-composite-value","seed":202611192043,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[20,1]],\"values\":[[17,1],[18,1],[20,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 20], aligned endpoint values=[17, 18, 20]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 20], [17, 18, 20], [35/2, 361], 757/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 20], [17, 18, 20], [35/2, 361], 757/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0044","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dacbc88901f84649f0b707c729711b69968dad5647cc3661e47b26f05255c4c","family_id":"qf-mod50-trap-composite-value","seed":202611192044,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[21,1]],\"values\":[[18,1],[19,1],[21,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 21], aligned endpoint values=[18, 19, 21]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 21], [18, 19, 21], [37/2, 400], 837/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 21], [18, 19, 21], [37/2, 400], 837/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0045","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce3e7144f25d71a3b8ae7cdd609014780b06717f934be689c12a2b434b3994b5","family_id":"qf-mod50-trap-composite-value","seed":202611192045,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[22,1]],\"values\":[[19,1],[20,1],[22,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 22], aligned endpoint values=[19, 20, 22]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 22], [19, 20, 22], [39/2, 441], 921/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 22], [19, 20, 22], [39/2, 441], 921/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0046","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"745608b6582bd38cfca15820c4e35949ecf229e62c8669575393ac4f9cd0e393","family_id":"qf-mod50-trap-composite-value","seed":202611192046,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[23,1]],\"values\":[[20,1],[21,1],[23,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 23], aligned endpoint values=[20, 21, 23]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 23], [20, 21, 23], [41/2, 484], 1009/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 23], [20, 21, 23], [41/2, 484], 1009/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0047","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48e2073ee58788d9094cba7d3b01f796198de3b223888a4c563da4665071c073","family_id":"qf-mod50-trap-composite-value","seed":202611192047,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[24,1]],\"values\":[[21,1],[22,1],[24,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 24], aligned endpoint values=[21, 22, 24]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 24], [21, 22, 24], [43/2, 529], 1101/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 24], [21, 22, 24], [43/2, 529], 1101/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0048","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b68c2ef5055afccd90ee524927ae9cf744874df477e950eb8cea765d95d7491","family_id":"qf-mod50-trap-composite-value","seed":202611192048,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[25,1]],\"values\":[[22,1],[23,1],[25,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 25], aligned endpoint values=[22, 23, 25]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 25], [22, 23, 25], [45/2, 576], 1197/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 25], [22, 23, 25], [45/2, 576], 1197/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0049","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ca1bae3a13c928d693c58777b702ff43f632759cac5c855e480b6b3a39f53a","family_id":"qf-mod50-trap-composite-value","seed":202611192049,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[26,1]],\"values\":[[23,1],[24,1],[26,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 26], aligned endpoint values=[23, 24, 26]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 26], [23, 24, 26], [47/2, 625], 1297/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 26], [23, 24, 26], [47/2, 625], 1297/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0050","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63bf02877760440246567a9bec7d532f6e529ac2c7b829c2cdf9724d805c4fe4","family_id":"qf-mod50-trap-composite-value","seed":202611192050,"semantic_template":"trap-composite-value|{\"nodes\":[[0,1],[1,1],[27,1]],\"values\":[[24,1],[25,1],[27,1]]}"},"fields":{"title":"Each Trapezoid Uses Its Own Width and Endpoint Average","stem":"Use strictly increasing nodes=[0, 1, 27], aligned endpoint values=[24, 25, 27]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).","answer":{"kind":"expression","value":"([0, 1, 27], [24, 25, 27], [49/2, 676], 1401/2, each_subinterval_uses_its_own_width)","explanation":"For each interval multiply its own width by the average of its two endpoint values, then add the exact terms. The table fixes this quadrature sum; without a specified full integrand it does not determine a unique exact integral. The exact result, retaining the requested order, is ([0, 1, 27], [24, 25, 27], [49/2, 676], 1401/2, each_subinterval_uses_its_own_width)."},"skills":["Each Trapezoid Uses Its Own Width and Endpoint Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0051","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40de8c60b05c289755bae8152019c8753a1db418684a87c5e1ebefd15aa5f058","family_id":"qf-mod50-trap-affine-exactness","seed":202611192051,"semantic_template":"trap-affine-exactness|{\"intercept\":[1,1],\"nodes\":[[0,1],[1,1],[3,1]],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-3, b=1 and strictly increasing nodes=[0, 1, 3]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-3, 1, [0, 1, 3], [1, -2, -8], -21/2, -21/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 1, [0, 1, 3], [1, -2, -8], -21/2, -21/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0052","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3158ddcbc3a0340439e7faedfebaf1679ff5dfe1adaeb10e1bff661152248bcd","family_id":"qf-mod50-trap-affine-exactness","seed":202611192052,"semantic_template":"trap-affine-exactness|{\"intercept\":[2,1],\"nodes\":[[0,1],[1,1],[4,1]],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-2, b=2 and strictly increasing nodes=[0, 1, 4]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-2, 2, [0, 1, 4], [2, 0, -6], -8, -8, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 2, [0, 1, 4], [2, 0, -6], -8, -8, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0053","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbbfaa29d25f1b48b86076924d1c18bada99cb1cd687752cc601d3a9d5fe6075","family_id":"qf-mod50-trap-affine-exactness","seed":202611192053,"semantic_template":"trap-affine-exactness|{\"intercept\":[3,1],\"nodes\":[[0,1],[1,1],[5,1]],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-1, b=3 and strictly increasing nodes=[0, 1, 5]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-1, 3, [0, 1, 5], [3, 2, -2], 5/2, 5/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 3, [0, 1, 5], [3, 2, -2], 5/2, 5/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0054","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffe15de52d21fe7781d5dee759c4e8fb9d7cf4104521bf793615f7dbd920c0a6","family_id":"qf-mod50-trap-affine-exactness","seed":202611192054,"semantic_template":"trap-affine-exactness|{\"intercept\":[4,1],\"nodes\":[[0,1],[1,1],[6,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=4 and strictly increasing nodes=[0, 1, 6]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 4, [0, 1, 6], [4, 6, 16], 60, 60, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 4, [0, 1, 6], [4, 6, 16], 60, 60, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0055","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ddd25605218fb7092d3a37a7e1db5afcb8f21161a746674b09f2ca7aeed2587","family_id":"qf-mod50-trap-affine-exactness","seed":202611192055,"semantic_template":"trap-affine-exactness|{\"intercept\":[5,1],\"nodes\":[[0,1],[1,1],[7,1]],\"slope\":[1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=1, b=5 and strictly increasing nodes=[0, 1, 7]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(1, 5, [0, 1, 7], [5, 6, 12], 119/2, 119/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 5, [0, 1, 7], [5, 6, 12], 119/2, 119/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0056","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7f2cb08fd9e6bb60a255e5bee9b2418f1aa2f1c94699396bb5a46f2ccc0d88d","family_id":"qf-mod50-trap-affine-exactness","seed":202611192056,"semantic_template":"trap-affine-exactness|{\"intercept\":[6,1],\"nodes\":[[0,1],[1,1],[8,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=6 and strictly increasing nodes=[0, 1, 8]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 6, [0, 1, 8], [6, 8, 22], 112, 112, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 6, [0, 1, 8], [6, 8, 22], 112, 112, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0057","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f6a29fcb72e17004fbef19ae0fb84d5a0377df35844d794f049dec4c4e2877","family_id":"qf-mod50-trap-affine-exactness","seed":202611192057,"semantic_template":"trap-affine-exactness|{\"intercept\":[7,1],\"nodes\":[[0,1],[1,1],[9,1]],\"slope\":[3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=3, b=7 and strictly increasing nodes=[0, 1, 9]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(3, 7, [0, 1, 9], [7, 10, 34], 369/2, 369/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 7, [0, 1, 9], [7, 10, 34], 369/2, 369/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0058","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba27a02f94a6a84fd76b53dfd343d613a7c4f8d3a92f78103681998a35d89be8","family_id":"qf-mod50-trap-affine-exactness","seed":202611192058,"semantic_template":"trap-affine-exactness|{\"intercept\":[8,1],\"nodes\":[[0,1],[1,1],[10,1]],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-3, b=8 and strictly increasing nodes=[0, 1, 10]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-3, 8, [0, 1, 10], [8, 5, -22], -70, -70, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 8, [0, 1, 10], [8, 5, -22], -70, -70, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0059","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17260127694e41396d234f7eb0dfbc6cdaeb32cebb379aa801a14a36b5e84e8d","family_id":"qf-mod50-trap-affine-exactness","seed":202611192059,"semantic_template":"trap-affine-exactness|{\"intercept\":[9,1],\"nodes\":[[0,1],[1,1],[11,1]],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-2, b=9 and strictly increasing nodes=[0, 1, 11]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-2, 9, [0, 1, 11], [9, 7, -13], -22, -22, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 9, [0, 1, 11], [9, 7, -13], -22, -22, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0060","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7440b4749b9ce69cf394753004a953ceca18116c87378fb730328aa8fdd1035","family_id":"qf-mod50-trap-affine-exactness","seed":202611192060,"semantic_template":"trap-affine-exactness|{\"intercept\":[10,1],\"nodes\":[[0,1],[1,1],[12,1]],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-1, b=10 and strictly increasing nodes=[0, 1, 12]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-1, 10, [0, 1, 12], [10, 9, -2], 48, 48, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 10, [0, 1, 12], [10, 9, -2], 48, 48, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0061","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a6341d60d5bac56ded4c8dd2a9c577f3e5484791e4e2cabe4f077c240a5e511","family_id":"qf-mod50-trap-affine-exactness","seed":202611192061,"semantic_template":"trap-affine-exactness|{\"intercept\":[11,1],\"nodes\":[[0,1],[1,1],[13,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=11 and strictly increasing nodes=[0, 1, 13]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 11, [0, 1, 13], [11, 13, 37], 312, 312, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 11, [0, 1, 13], [11, 13, 37], 312, 312, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0062","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a36cc73dd4868ea3e9b3f6b16b6f2ccd7f6afa2bf8054ce0a96d658d19d071","family_id":"qf-mod50-trap-affine-exactness","seed":202611192062,"semantic_template":"trap-affine-exactness|{\"intercept\":[12,1],\"nodes\":[[0,1],[1,1],[14,1]],\"slope\":[1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=1, b=12 and strictly increasing nodes=[0, 1, 14]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(1, 12, [0, 1, 14], [12, 13, 26], 266, 266, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 12, [0, 1, 14], [12, 13, 26], 266, 266, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0063","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536fda819cd36d13fa4773e226ebfbf49d2349e6b3fecde657bce69199d541dd","family_id":"qf-mod50-trap-affine-exactness","seed":202611192063,"semantic_template":"trap-affine-exactness|{\"intercept\":[13,1],\"nodes\":[[0,1],[1,1],[15,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=13 and strictly increasing nodes=[0, 1, 15]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 13, [0, 1, 15], [13, 15, 43], 420, 420, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 13, [0, 1, 15], [13, 15, 43], 420, 420, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0064","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ec6eefc869df1ed0f0bb56c1d6db6a93809b92f068eba6344c489f2c8542b6","family_id":"qf-mod50-trap-affine-exactness","seed":202611192064,"semantic_template":"trap-affine-exactness|{\"intercept\":[14,1],\"nodes\":[[0,1],[1,1],[16,1]],\"slope\":[3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=3, b=14 and strictly increasing nodes=[0, 1, 16]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(3, 14, [0, 1, 16], [14, 17, 62], 608, 608, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 14, [0, 1, 16], [14, 17, 62], 608, 608, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0065","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c330e8c804b7dcc7aaa5211811964047cd219c87591121ddac4805990a4023c5","family_id":"qf-mod50-trap-affine-exactness","seed":202611192065,"semantic_template":"trap-affine-exactness|{\"intercept\":[15,1],\"nodes\":[[0,1],[1,1],[17,1]],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-3, b=15 and strictly increasing nodes=[0, 1, 17]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-3, 15, [0, 1, 17], [15, 12, -36], -357/2, -357/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 15, [0, 1, 17], [15, 12, -36], -357/2, -357/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0066","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"584db5b5964d726d0f32656a37452d30ca9a42f49c044a076139cfb4f625d909","family_id":"qf-mod50-trap-affine-exactness","seed":202611192066,"semantic_template":"trap-affine-exactness|{\"intercept\":[16,1],\"nodes\":[[0,1],[1,1],[18,1]],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-2, b=16 and strictly increasing nodes=[0, 1, 18]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-2, 16, [0, 1, 18], [16, 14, -20], -36, -36, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 16, [0, 1, 18], [16, 14, -20], -36, -36, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0067","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c79a63d0be6d5ca99718865d8dde84154d033381aaa515f3f2d07d9fe5855691","family_id":"qf-mod50-trap-affine-exactness","seed":202611192067,"semantic_template":"trap-affine-exactness|{\"intercept\":[17,1],\"nodes\":[[0,1],[1,1],[19,1]],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-1, b=17 and strictly increasing nodes=[0, 1, 19]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-1, 17, [0, 1, 19], [17, 16, -2], 285/2, 285/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 17, [0, 1, 19], [17, 16, -2], 285/2, 285/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0068","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ed9fdc16e5441ecc4beb5a7f708b9752d0c0f814afea0a90bbf4ea6bc6dcc2","family_id":"qf-mod50-trap-affine-exactness","seed":202611192068,"semantic_template":"trap-affine-exactness|{\"intercept\":[18,1],\"nodes\":[[0,1],[1,1],[20,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=18 and strictly increasing nodes=[0, 1, 20]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 18, [0, 1, 20], [18, 20, 58], 760, 760, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 18, [0, 1, 20], [18, 20, 58], 760, 760, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0069","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f003fca674502480223e32172990861a2982a26225ad589f6e1d51b7c887bf5","family_id":"qf-mod50-trap-affine-exactness","seed":202611192069,"semantic_template":"trap-affine-exactness|{\"intercept\":[19,1],\"nodes\":[[0,1],[1,1],[21,1]],\"slope\":[1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=1, b=19 and strictly increasing nodes=[0, 1, 21]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(1, 19, [0, 1, 21], [19, 20, 40], 1239/2, 1239/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (1, 19, [0, 1, 21], [19, 20, 40], 1239/2, 1239/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0070","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e01689ffeee6ea0bd2eea83c4f1eab5bf454a762d8d4814920229cc1279bc72","family_id":"qf-mod50-trap-affine-exactness","seed":202611192070,"semantic_template":"trap-affine-exactness|{\"intercept\":[20,1],\"nodes\":[[0,1],[1,1],[22,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=20 and strictly increasing nodes=[0, 1, 22]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 20, [0, 1, 22], [20, 22, 64], 924, 924, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 20, [0, 1, 22], [20, 22, 64], 924, 924, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0071","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9af17d18ba104b1b6345a9a28d05eae406d75283e112e4e1dfa1daeaf4756a26","family_id":"qf-mod50-trap-affine-exactness","seed":202611192071,"semantic_template":"trap-affine-exactness|{\"intercept\":[21,1],\"nodes\":[[0,1],[1,1],[23,1]],\"slope\":[3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=3, b=21 and strictly increasing nodes=[0, 1, 23]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(3, 21, [0, 1, 23], [21, 24, 90], 2553/2, 2553/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (3, 21, [0, 1, 23], [21, 24, 90], 2553/2, 2553/2, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0072","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"270f65c4db1c88d65d17bb945b74d59e13907dad89a9c218475808520a47a913","family_id":"qf-mod50-trap-affine-exactness","seed":202611192072,"semantic_template":"trap-affine-exactness|{\"intercept\":[22,1],\"nodes\":[[0,1],[1,1],[24,1]],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-3, b=22 and strictly increasing nodes=[0, 1, 24]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-3, 22, [0, 1, 24], [22, 19, -50], -336, -336, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-3, 22, [0, 1, 24], [22, 19, -50], -336, -336, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0073","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5138ae3f45f18c61d9899f5dfb29470c2ee027e67db49f820901a20904199ffd","family_id":"qf-mod50-trap-affine-exactness","seed":202611192073,"semantic_template":"trap-affine-exactness|{\"intercept\":[23,1],\"nodes\":[[0,1],[1,1],[25,1]],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-2, b=23 and strictly increasing nodes=[0, 1, 25]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-2, 23, [0, 1, 25], [23, 21, -27], -50, -50, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-2, 23, [0, 1, 25], [23, 21, -27], -50, -50, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0074","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969acb21b698546e291d3126b2e73bf3db4fc8a7668d69cbaccd9d8303c76dd1","family_id":"qf-mod50-trap-affine-exactness","seed":202611192074,"semantic_template":"trap-affine-exactness|{\"intercept\":[24,1],\"nodes\":[[0,1],[1,1],[26,1]],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=-1, b=24 and strictly increasing nodes=[0, 1, 26]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(-1, 24, [0, 1, 26], [24, 23, -2], 286, 286, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (-1, 24, [0, 1, 26], [24, 23, -2], 286, 286, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0075","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0e4efc54213b59e199174f528b4ca7300a61d7694557cdd2fdcbc75160d70b6","family_id":"qf-mod50-trap-affine-exactness","seed":202611192075,"semantic_template":"trap-affine-exactness|{\"intercept\":[25,1],\"nodes\":[[0,1],[1,1],[27,1]],\"slope\":[2,1]}"},"fields":{"title":"An Affine Integrand Is Integrated Exactly on Any Qualified Partition","stem":"Use f(x)=m*x+b with m=2, b=25 and strictly increasing nodes=[0, 1, 27]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope).","answer":{"kind":"expression","value":"(2, 25, [0, 1, 27], [25, 27, 79], 1404, 1404, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)","explanation":"Calculate all endpoint values, the individual-width trapezoid sum and the polynomial antiderivative difference. They agree for this affine function on this partition. This does not imply exactness for arbitrary integrands or a general error-order theorem. The exact result, retaining the requested order, is (2, 25, [0, 1, 27], [25, 27, 79], 1404, 1404, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)."},"skills":["An Affine Integrand Is Integrated Exactly on Any Qualified Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r02-0076","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f68616a2c13245486cedd2f1fe613e96d651dc974328f9a7a42da8f502fe85f5","family_id":"qf-mod50-trap-error-boundary","seed":202611192076,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[1,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[3,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[1,1,1],\"nodes\":[0,1,3]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [1, 1, 1], [0, 1, 3], 18, 33/2, -3/2, 3/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [1, 1, 1], [0, 1, 3], 18, 33/2, -3/2, 3/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0077","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf44dec35d441fba1ce2aef1833499d176eeaa68675f5d0ee6fb67e7b526663c","family_id":"qf-mod50-trap-error-boundary","seed":202611192077,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[2,1]],\"values\":[[1,1],[2,1],[3,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,2],\"values\":[1,2,3]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 2], [1, 2, 3], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 2], [1, 2, 3], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0078","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7331db3a4dee2f653bdb3b52f453473c7c8c779abbb42b2b3b3e570b66a7c06f","family_id":"qf-mod50-trap-error-boundary","seed":202611192078,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[5,1]],\"values\":[[2,1],[3,1],[5,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,5],\"values\":[2,3,5]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 5], [2, 3, 5], 37/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 5], [2, 3, 5], 37/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0079","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef917596a514a65b62caddadd1f6b529c2560b16d5fe044b9cf83835ed332fa9","family_id":"qf-mod50-trap-error-boundary","seed":202611192079,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[6,1]],\"values\":[[3,1],[7,1],[13,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,6],\"values\":[3,7,13]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 5], 55, 15, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 5], 55, 15, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0080","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4181389fa0892ed12f23600312225f9e48fd10dcc81661d9f7919ccae9ee7164","family_id":"qf-mod50-trap-error-boundary","seed":202611192080,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[5,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[7,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[5,1,1],\"nodes\":[0,1,7]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [5, 1, 1], [0, 1, 7], 210, 1043/6, -217/6, 217/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [5, 1, 1], [0, 1, 7], 210, 1043/6, -217/6, 217/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0081","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e2ddc8e7d051eacf7dba425be327cbbe5f0d057acdce5002884d740ea3f0633","family_id":"qf-mod50-trap-error-boundary","seed":202611192081,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[6,1]],\"values\":[[5,1],[6,1],[7,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,6],\"values\":[5,6,7]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 6], [5, 6, 7], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 6], [5, 6, 7], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0082","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56343c04868ed855e288e0954f5982df0aa70382105429bd3c35b92947595da8","family_id":"qf-mod50-trap-error-boundary","seed":202611192082,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[9,1]],\"values\":[[6,1],[7,1],[9,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,9],\"values\":[6,7,9]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 9], [6, 7, 9], 141/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 9], [6, 7, 9], 141/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0083","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9bf441a40e072a0a92de14507e334c76b17f52ac722225290b8c20bbb0a62c","family_id":"qf-mod50-trap-error-boundary","seed":202611192083,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[10,1]],\"values\":[[7,1],[15,1],[25,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,10],\"values\":[7,15,25]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 9], 191, 31, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 9], 191, 31, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0084","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d9a0c130567adc0d25dc16769d375db33398926f2c1b7553556872bf3d71e7b","family_id":"qf-mod50-trap-error-boundary","seed":202611192084,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[9,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[11,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[9,1,1],\"nodes\":[0,1,11]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [9, 1, 1], [0, 1, 11], 770, 3619/6, -1001/6, 1001/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [9, 1, 1], [0, 1, 11], 770, 3619/6, -1001/6, 1001/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0085","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65ab130897ebe7b1c6fd06386ab8c0f784c8028b3b61b0c5edb4be93360c82a3","family_id":"qf-mod50-trap-error-boundary","seed":202611192085,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[10,1]],\"values\":[[9,1],[10,1],[11,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,10],\"values\":[9,10,11]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 10], [9, 10, 11], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 10], [9, 10, 11], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0086","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c161ee959a153e45f6576cef6c17c36dfaa38f0bed91e4aa7986755be59511e","family_id":"qf-mod50-trap-error-boundary","seed":202611192086,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[13,1]],\"values\":[[10,1],[11,1],[13,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,13],\"values\":[10,11,13]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 13], [10, 11, 13], 309/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 13], [10, 11, 13], 309/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0087","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6085a6cc43a14391fb988c03d320d268c65c5c96cc63e0da352d83aaaf28588","family_id":"qf-mod50-trap-error-boundary","seed":202611192087,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[14,1]],\"values\":[[11,1],[23,1],[37,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,14],\"values\":[11,23,37]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 13], 407, 47, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 13], 407, 47, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0088","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32860fa1cf4a30bf7ce696fc9d78eb7d6337b5ba63f9e419f16eb871d7bc2947","family_id":"qf-mod50-trap-error-boundary","seed":202611192088,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[13,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[15,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[13,1,1],\"nodes\":[0,1,15]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [13, 1, 1], [0, 1, 15], 1890, 2865/2, -915/2, 915/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [13, 1, 1], [0, 1, 15], 1890, 2865/2, -915/2, 915/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0089","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8251d7e7443305ee3e40d3e5b31fc16bb9287c4143dd490c1d86264dc4bc640e","family_id":"qf-mod50-trap-error-boundary","seed":202611192089,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[14,1]],\"values\":[[13,1],[14,1],[15,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,14],\"values\":[13,14,15]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 14], [13, 14, 15], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 14], [13, 14, 15], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0090","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0e4a537397db7bf86de1779d00abf0eb0257c4e3dd829abd61721b2c16e0fd7","family_id":"qf-mod50-trap-error-boundary","seed":202611192090,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[17,1]],\"values\":[[14,1],[15,1],[17,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,17],\"values\":[14,15,17]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 17], [14, 15, 17], 541/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 17], [14, 15, 17], 541/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0091","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fe991738f52b38d3155bfa6fd5320e1e9ede3d1bba7ee69025ece9cc33afc0e","family_id":"qf-mod50-trap-error-boundary","seed":202611192091,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[18,1]],\"values\":[[15,1],[31,1],[49,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,18],\"values\":[15,31,49]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 17], 703, 63, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 17], 703, 63, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0092","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c43b15304d369781a5878698ac7e7d57c58b98a49ae7c7026bcbed75e0d03876","family_id":"qf-mod50-trap-error-boundary","seed":202611192092,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[17,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[19,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[17,1,1],\"nodes\":[0,1,19]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [17, 1, 1], [0, 1, 19], 3762, 16739/6, -5833/6, 5833/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0093","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c11cb340553d768c11aecf86f6bbc4f74c8d438addb0d16bda291950010a08e","family_id":"qf-mod50-trap-error-boundary","seed":202611192093,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[18,1]],\"values\":[[17,1],[18,1],[19,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,18],\"values\":[17,18,19]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 18], [17, 18, 19], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 18], [17, 18, 19], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0094","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15e1ac01c09a3360d6556c973c1ab62abaf50aee432b4a264024300bb5e32b9c","family_id":"qf-mod50-trap-error-boundary","seed":202611192094,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[21,1]],\"values\":[[18,1],[19,1],[21,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,21],\"values\":[18,19,21]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 21], [18, 19, 21], 837/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 21], [18, 19, 21], 837/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0095","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"552fefc6539cec12e87389d661a18654e3eefb8a82b45c76618d18908430fd30","family_id":"qf-mod50-trap-error-boundary","seed":202611192095,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[22,1]],\"values\":[[19,1],[39,1],[61,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,22],\"values\":[19,39,61]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 21], 1079, 79, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 21], 1079, 79, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0096","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da0b7288787cedc7827e635e68b05496a858858e8771a2ee9a103c8bf515a4e","family_id":"qf-mod50-trap-error-boundary","seed":202611192096,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[21,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[23,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[21,1,1],\"nodes\":[0,1,23]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [21, 1, 1], [0, 1, 23], 6578, 28819/6, -10649/6, 10649/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [21, 1, 1], [0, 1, 23], 6578, 28819/6, -10649/6, 10649/6, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0097","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f96e500acf67577338b6139a6fa241c3c34ec8fdedf562aab5ffae3a9322842c","family_id":"qf-mod50-trap-error-boundary","seed":202611192097,"semantic_template":"trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[22,1]],\"values\":[[21,1],[22,1],[23,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=nonincreasing_partition, complete model={\"nodes\":[0,0,22],\"values\":[21,22,23]}. Give (mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary).","answer":{"kind":"expression","value":"(nonincreasing_partition, [0, 0, 22], [21, 22, 23], Yes, No, No, partition_must_be_strictly_increasing)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (nonincreasing_partition, [0, 0, 22], [21, 22, 23], Yes, No, No, partition_must_be_strictly_increasing)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0098","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63cad8e9ee9761287b56158dc057f6494550c67a4eb558136ef24fab32cbdeac","family_id":"qf-mod50-trap-error-boundary","seed":202611192098,"semantic_template":"trap-error-boundary|{\"mode\":\"sample_table_scope\",\"model\":{\"nodes\":[[0,1],[1,1],[25,1]],\"values\":[[22,1],[23,1],[25,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=sample_table_scope, complete model={\"nodes\":[0,1,25],\"values\":[22,23,25]}. Give (mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary).","answer":{"kind":"expression","value":"(sample_table_scope, [0, 1, 25], [22, 23, 25], 1197/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (sample_table_scope, [0, 1, 25], [22, 23, 25], 1197/2, Yes, No, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0099","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"165a3551ab38898ad87cec2869bf5fb557446dc1bfd00dccebfd2092dcf6b6c9","family_id":"qf-mod50-trap-error-boundary","seed":202611192099,"semantic_template":"trap-error-boundary|{\"mode\":\"single_width_confusion\",\"model\":{\"nodes\":[[0,1],[1,1],[26,1]],\"values\":[[23,1],[47,1],[73,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=single_width_confusion, complete model={\"nodes\":[0,1,26],\"values\":[23,47,73]}. Give (mode, individual_widths, correct_value, wrong_single_width_value, single_width_method_claim, boundary).","answer":{"kind":"expression","value":"(single_width_confusion, [1, 25], 1535, 95, No, nonuniform_partition_cannot_use_one_global_width)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (single_width_confusion, [1, 25], 1535, 95, No, nonuniform_partition_cannot_use_one_global_width)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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-pulse-20261119-r02-0100","batch_id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040718e3f916e9130794c08f7b4cd4ca8985d5c6d3595582a98236d1801f8ca0","family_id":"qf-mod50-trap-error-boundary","seed":202611192100,"semantic_template":"trap-error-boundary|{\"mode\":\"quadratic_reference\",\"model\":{\"coefficients\":[[25,1],[1,1],[1,1]],\"nodes\":[[0,1],[1,1],[27,1]]}}"},"fields":{"title":"Quadratic Reference Error Is Not a General Quadrature Error Theorem","stem":"Use mode=quadratic_reference, complete model={\"coefficients\":[25,1,1],\"nodes\":[0,1,27]}. Give (mode, coefficients_low_to_high, nodes, trapezoid_value, exact_integral, signed_error_I_minus_T, absolute_error, general_error_bound_claim, scope).","answer":{"kind":"expression","value":"(quadratic_reference, [25, 1, 1], [0, 1, 27], 10530, 15201/2, -5859/2, 5859/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)","explanation":"The full quadratic polynomial permits an exact reference integral and signed error I−T, with absolute error separate. A nonincreasing node list is invalid and not repaired by sorting. Endpoint samples alone do not specify the integrand or exact integral. A discriminating nonuniform example refutes using the first interval width globally. The exact result, retaining the requested order, is (quadratic_reference, [25, 1, 1], [0, 1, 27], 10530, 15201/2, -5859/2, 5859/2, No, exact_polynomial_reference_error_is_not_a_general_second_derivative_error_bound)."},"skills":["Quadratic Reference Error Is Not a General Quadrature Error 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."}}