{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r02-0085",
  "pilot_number": "R0055-28160",
  "number": 28160,
  "family_id": "qf-mod50-trap-error-boundary",
  "title": {
    "zh-Hant": "第 28160 題：複合梯形法",
    "en": "Question 28160: Quadratic Reference Error Is Not a General Quadrature Error Theorem"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r02-0085",
    "batch_id": "batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-trapezoidal-quadrature"
    ],
    "title_zh": "複合梯形法",
    "stem_zh": "嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[0,0,10],\"values\":[9,10,11]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(nonincreasing_partition, [0, 0, 10], [9, 10, 11], 是, 否, 否, partition_must_be_strictly_increasing)",
      "explanation_zh": "Wave6第十三層核對=(nonincreasing_partition, [0, 0, 10], [9, 10, 11], 是, 否, 否, partition_must_be_strictly_increasing)；依完整明示資料精確重算；複合梯形逐段使用endpoint average與自己的width；signed error固定為I-T；affine exact或指定quadratic reference不推出一般誤差界、收斂階或唯一integrand。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "複合梯形法"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-trap-error-boundary",
      "seed": 202611192085,
      "parameters": {
        "mode": "nonincreasing_partition",
        "model": {
          "nodes": [
            0,
            0,
            10
          ],
          "values": [
            9,
            10,
            11
          ]
        },
        "domain": "Exact_composite_trapezoidal_quadrature",
        "operation": "trap-error-boundary",
        "semantic_template": "trap-error-boundary|{\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[[0,1],[0,1],[10,1]],\"values\":[[9,1],[10,1],[11,1]]}}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_contract_recomputation",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "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."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={\"mode\":\"nonincreasing_partition\",\"model\":{\"nodes\":[0,0,10],\"values\":[9,10,11]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "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)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-trap-error-boundary, trap-error-boundary]",
      "after": "derived = [[mode, nodes, endpoint_values, aligned, strictly_increasing, partition_qualification_claim, boundary]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[0, 0, 10], [9, 10, 11]]",
      "after": "derived = [是, 否, 否]",
      "relation": "implies",
      "rule_id": "Trapezoid_ACTUAL_aligned_nonincreasing_partition_not_repaired_by_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這個boundary只驗aligned且非strict increase的真見證；不排序修復，也不套normal partition的24-node界或計不合格width sums。",
        "en": "This boundary requires an actually aligned nonincreasing witness.Do not repair it by sorting,import the normal24-node cap or calculate an unqualified quadrature."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "nonincreasing_partition",
      [
        "0",
        "0",
        "10"
      ],
      [
        "9",
        "10",
        "11"
      ],
      true,
      false,
      false,
      "partition_must_be_strictly_increasing"
    ]
  },
  "conclusion": {
    "canonical_value": "(nonincreasing_partition, [0, 0, 10], [9, 10, 11], 是, 否, 否, partition_must_be_strictly_increasing)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/questions.jsonl",
      "batch_sha256": "42c2a2253cda785cc89b1d9f6e2ca25dc3eedad80f592c69c8f2637ae2511f18",
      "original_line": 85,
      "original_line_sha256": "f9f8653e1772049cccb8f2f419c2b17ebdda7dc34337051f6164ee5aff304599",
      "question_compact_sha256": "65ab130897ebe7b1c6fd06386ab8c0f784c8028b3b61b0c5edb4be93360c82a3",
      "english_row_compact_sha256": "6a1cc8f9d0928f63199b1c1a0abc33d8d6a22f509fb6cd729931364d3d506f42",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-trapezoidal-quadrature": {
        "sha256": "bd70a77af9c5b1fc3dbca7b29f06de4dbb579be5a6c48bbb16df6141f4f90001",
        "english_sha256": "00cf21dfab1295bd666c4883317d8bf1a960f6d5c2020d9b82fa4147c1779e80",
        "original_path": "public/data/mko/trapezoidal-quadrature.json",
        "english_path": "public/data/i18n/editions/mko-trapezoidal-quadrature.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "58cc90316e8f6898cdeb2a1adac25808042bc4b89f4faad89cf99130e15e6a7b",
    "original_candidate_line_sha256": "58cc90316e8f6898cdeb2a1adac25808042bc4b89f4faad89cf99130e15e6a7b"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p56",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "11728aa004e6c2bfd116b1c75a834088785806b2003aecab93db94a6a885f437",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}