{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r05-0040",
  "pilot_number": "R0066-33940",
  "number": 33940,
  "family_id": "qf-mod50-numerr-affine-propagation",
  "title": {
    "zh-Hant": "第 33940 題：數值近似誤差",
    "en": "Question 33940: An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r05-0040",
    "batch_id": "batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-numerical-approximation-error"
    ],
    "title_zh": "數值近似誤差",
    "stem_zh": "exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[15,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(-3, 5, 15, affine_output_absolute_error_bound)",
      "explanation_zh": "Wave6第十三層核對=(-3, 5, 15, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "數值近似誤差"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-numerr-affine-propagation",
      "seed": 202611195040,
      "parameters": {
        "slope": -3,
        "input_error_bound": [
          15,
          3
        ],
        "domain": "Exact_numerical_error_density",
        "operation": "numerr-affine-propagation",
        "semantic_template": "numerr-affine-propagation|{\"input_error_bound\":[5,1],\"slope\":[-3,1]}"
      }
    },
    "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-r05-0040",
    "batch_id": "batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1dc93cc20f7f11d7b9a41d9da880c21169f5111f19eaf99457946e1ff84266ea",
      "family_id": "qf-mod50-numerr-affine-propagation",
      "seed": 202611195040,
      "semantic_template": "numerr-affine-propagation|{\"input_error_bound\":[5,1],\"slope\":[-3,1]}"
    },
    "fields": {
      "title": "An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope",
      "stem": "Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).",
      "answer": {
        "kind": "expression",
        "value": "(-3, 5, 15, affine_output_absolute_error_bound)",
        "explanation": "Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 5, 15, affine_output_absolute_error_bound)."
      },
      "skills": [
        "An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"
      ],
      "verification_warning": "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": "exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[15,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-numerr-affine-propagation, [domain, input_error_bound, operation, slope]]",
      "after": "derived = [numerr-affine-propagation, Exact_numerical_error_density]",
      "relation": "implies",
      "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。",
        "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [propagation ORIGINAL slope, 100000000, -3, 1]",
      "after": "derived = [[-3, 1], -3]",
      "relation": "implies",
      "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。",
        "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "propagation ORIGINAL slope ORIGINAL numerator over denominator := div(-3, 1)",
      "after": "propagation ORIGINAL slope ORIGINAL numerator over denominator = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 propagation ORIGINAL slope ORIGINAL numerator over denominator，不以小數近似替代。",
          "en": "Recompute propagation ORIGINAL slope ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "propagation ORIGINAL slope ORIGINAL numerator over denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [propagation ORIGINAL input bound, 100000000, 15, 3]",
      "after": "derived = [[5, 1], 5]",
      "relation": "implies",
      "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。",
        "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "propagation ORIGINAL input bound ORIGINAL numerator over denominator := div(15, 3)",
      "after": "propagation ORIGINAL input bound ORIGINAL numerator over denominator = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 propagation ORIGINAL input bound ORIGINAL numerator over denominator，不以小數近似替代。",
          "en": "Recompute propagation ORIGINAL input bound ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "3"
        ],
        "output": "5",
        "label": "propagation ORIGINAL input bound ORIGINAL numerator over denominator"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Affine output error bound SOURCE naive mul, 100000000, 15, 1]",
      "after": "derived = [[15, 1], 15]",
      "relation": "implies",
      "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。",
        "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator := div(15, 1)",
      "after": "Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。",
          "en": "Recompute Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "1"
        ],
        "output": "15",
        "label": "Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Affine output error bound ORIGINAL rational mul := mul(3, 5)",
      "after": "Affine output error bound ORIGINAL rational mul = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine output error bound ORIGINAL rational mul，不以小數近似替代。",
          "en": "Recompute Affine output error bound ORIGINAL rational mul with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "5"
        ],
        "output": "15",
        "label": "Affine output error bound ORIGINAL rational mul"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [-3, 5]",
      "after": "derived = [15]",
      "relation": "implies",
      "rule_id": "Error_AFFINE_propagation_absolute_bound_INCLUDES_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由|m(x+δ)+b−(mx+b)|=|m|*|δ|≤|m|*bound；bound非負，m或bound為0合法。這是此affine model的absolute bound，不辨識rounding/truncation/measurement來源或證convergence。",
        "en": "Use |m(x+δ)+b−(mx+b)|=|m|*|δ|≤|m|*bound.The bound is nonnegative;zero slope or bound is allowed.This is THIS affine model’s absolute bound,not identification of rounding/truncation/measurement sources or a convergence proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-3",
      "5",
      "15",
      "affine_output_absolute_error_bound"
    ]
  },
  "conclusion": {
    "canonical_value": "(-3, 5, 15, affine_output_absolute_error_bound)",
    "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-r05-numerical-approximation-error-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations/questions.jsonl",
      "batch_sha256": "932ee14b8b2a8fed6d8cebd6df3641eed4560176b36945b28664b2208604fe14",
      "original_line": 40,
      "original_line_sha256": "de283a407e7fa6f6a51d5f01b6cef48aaa4ceaccad59f85af14d51480bebf31f",
      "question_compact_sha256": "1dc93cc20f7f11d7b9a41d9da880c21169f5111f19eaf99457946e1ff84266ea",
      "english_row_compact_sha256": "3f7f23ece130843bd584847831b5ca6d68b0f3b4675ce6c2fd0fc1c8d51cda68",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "english_sha256": "fe552b17f866ceee13097202f2ac4dd54a72ef64059402444b22146b41d82149",
        "original_path": "public/data/mko/numerical-approximation-error.json",
        "english_path": "public/data/i18n/editions/mko-numerical-approximation-error.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310",
    "original_candidate_sha256": "c10a597de3384f84748b42627c9d76c96a2fb026b02f5b7e7254f50620b30c9d",
    "original_candidate_line_sha256": "c10a597de3384f84748b42627c9d76c96a2fb026b02f5b7e7254f50620b30c9d"
  },
  "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-20261026-012-p67",
    "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": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267",
      "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "66828fcc8ab84d284b7a46e9e3299e0194d374a402da9efed6596050daaa9f67",
      "arithmetic_steps": 4
    },
    "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."
  }
}