{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260930-r04-composition-order-comparison-0011",
  "pilot_number": "R0005-03086",
  "number": 3086,
  "family_id": "qf-composition-order-comparison",
  "title": {
    "zh-Hant": "第 3086 題：兩種合成順序比較",
    "en": "Question 3086: Comparing Two Composition Orders"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260930-r04-composition-order-comparison-0011",
    "batch_id": "batch-20260930-r04-composition-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-composition"
    ],
    "title_zh": "兩種合成順序比較",
    "stem_zh": "明示總函數 f:R→R，f(t)=(1)t+(-2)；g:R→R，g(t)=(1)t+(-4)。在 x=-2 時，(g∘f)(x) 與 (f∘g)(x) 是否相等？只比較此明示輸入。",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "兩種順序的值分別為 -8, -8；所以答案為是。此題不是由單一點證明任意函數交換。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "求值順序",
        "逐輸入比較"
      ]
    },
    "variant": {
      "family_id": "qf-composition-order-comparison",
      "seed": 202609304311,
      "parameters": {
        "operation": "compare_at_input",
        "a": 1,
        "b": -2,
        "c": 1,
        "d": -4,
        "x": -2,
        "semantic_template": "qf-composition-order-comparison/{\"a\":1,\"b\":-2,\"c\":1,\"d\":-4,\"operation\":\"compare_at_input\",\"x\":-2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "json_schema",
        "producer_local_answer_invariants",
        "explanation_witness_accuracy"
      ],
      "warning_zh": "本題保持 candidate；有限精確計算與產生者自檢不構成人工審查、形式證明或主架構者驗收。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex GPT-6 question producer",
      "generated_at": "2026-09-30",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260930-r04-composition-order-comparison-0011",
    "batch_id": "batch-20260930-r04-composition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "439f9b684ce5cf1422dc9361073433c859392be8499b80eb7ae727e94b8cc724",
      "family_id": "qf-composition-order-comparison",
      "seed": 202609304311,
      "semantic_template": "qf-composition-order-comparison/{\"a\":1,\"b\":-2,\"c\":1,\"d\":-4,\"operation\":\"compare_at_input\",\"x\":-2}"
    },
    "fields": {
      "title": "Comparing Two Composition Orders",
      "stem": "Let f(t)=(1)t+(-2) and g(t)=(1)t+(-4). At the explicit input x=-2, are (g∘f)(x) and (f∘g)(x) equal?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The two values are -8 and -8, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim."
      },
      "skills": [
        "evaluation order",
        "input-by-input comparison"
      ],
      "verification_warning": "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": "明示總函數 f:R→R，f(t)=(1)t+(-2)；g:R→R，g(t)=(1)t+(-4)。在 x=-2 時，(g∘f)(x) 與 (f∘g)(x) 是否相等？只比較此明示輸入。",
        "en": "Let f(t)=(1)t+(-2) and g(t)=(1)t+(-4). At the explicit input x=-2, are (g∘f)(x) and (f∘g)(x) equal?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "(g∘f)(x)",
      "after": "g(f(x)):apply f first,then g",
      "relation": "implies",
      "rule_id": "composition_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合成符號的右邊函數先作用，不是把兩個函數值相乘。",
        "en": "The function on the right acts first;composition is not multiplication of two function values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "f(x) product := mul(1, -2)",
      "after": "f(x) product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整代入目前輸入並乘係數。",
        "en": "Substitute the current input and multiply by the coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) product，不以小數近似替代。",
          "en": "Recompute f(x) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "f(x) product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "f(x) value := sum(-2, -2)",
      "after": "f(x) value = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此函數自己的常數項。",
        "en": "Then add this function's own constant term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) value，不以小數近似替代。",
          "en": "Recompute f(x) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "-4",
        "label": "f(x) value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "g(f(x)) product := mul(1, -4)",
      "after": "g(f(x)) product = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整代入目前輸入並乘係數。",
        "en": "Substitute the current input and multiply by the coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(f(x)) product，不以小數近似替代。",
          "en": "Recompute g(f(x)) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-4"
        ],
        "output": "-4",
        "label": "g(f(x)) product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "g(f(x)) value := sum(-4, -4)",
      "after": "g(f(x)) value = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此函數自己的常數項。",
        "en": "Then add this function's own constant term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(f(x)) value，不以小數近似替代。",
          "en": "Recompute g(f(x)) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "-4"
        ],
        "output": "-8",
        "label": "g(f(x)) value"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "g(x) product := mul(1, -2)",
      "after": "g(x) product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整代入目前輸入並乘係數。",
        "en": "Substitute the current input and multiply by the coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) product，不以小數近似替代。",
          "en": "Recompute g(x) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "g(x) product"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "g(x) value := sum(-2, -4)",
      "after": "g(x) value = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此函數自己的常數項。",
        "en": "Then add this function's own constant term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) value，不以小數近似替代。",
          "en": "Recompute g(x) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-4"
        ],
        "output": "-6",
        "label": "g(x) value"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "f(g(x)) product := mul(1, -6)",
      "after": "f(g(x)) product = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整代入目前輸入並乘係數。",
        "en": "Substitute the current input and multiply by the coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(g(x)) product，不以小數近似替代。",
          "en": "Recompute f(g(x)) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-6"
        ],
        "output": "-6",
        "label": "f(g(x)) product"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "f(g(x)) value := sum(-6, -2)",
      "after": "f(g(x)) value = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此函數自己的常數項。",
        "en": "Then add this function's own constant term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(g(x)) value，不以小數近似替代。",
          "en": "Recompute f(g(x)) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "-2"
        ],
        "output": "-8",
        "label": "f(g(x)) value"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "g(f(-2))=-8;f(g(-2))=-8",
      "after": "equal at supplied input=是",
      "relation": "implies",
      "rule_id": "pointwise_composition_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較此明示輸入；不能由任意兩函數在一點相等推論處處交換。本仿射模型若另推導係數恆等式，才可單獨討論全域性。",
        "en": "Compare the supplied input only;one-point equality of arbitrary functions does not prove commutation everywhere.A separate affine coefficient identity could address this particular model globally."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": true,
    "g_after_f": "-8",
    "f_after_g": "-8",
    "claim_scope": "supplied_input"
  },
  "conclusion": {
    "canonical_value": "是",
    "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-20260930-r04-composition-foundations",
      "batch_path": "public/data/questions/batches/batch-20260930-r04-composition-foundations/questions.jsonl",
      "batch_sha256": "36c06ad1401a2402e83fc4db2843422ed626fdaa20ea488f30a0832aaf4c781f",
      "original_line": 86,
      "original_line_sha256": "624f00194543e8662c7ec0018c7c3058452eef2acf47558793112fb3b446a528",
      "question_compact_sha256": "439f9b684ce5cf1422dc9361073433c859392be8499b80eb7ae727e94b8cc724",
      "english_row_compact_sha256": "b26f28f9e9f1c7c1a3b72067ba1199e7c83f607af63c1a9dbf2b29080697fd09",
      "english_manifest_path": "public/data/i18n/editions/batch-20260930-r04-composition-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260930-r04-composition-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-composition": {
        "sha256": "a915b980c52ed5cfa4fd9ef59e2a23ef237a0eebbe262cd6f1a67b15f052bfa9",
        "english_sha256": "702841fe3d6cacc8f4b48338fbd4fc96731540292253b3030c43dce8e1cf1979",
        "original_path": "public/data/mko/composition.json",
        "english_path": "public/data/i18n/editions/mko-composition.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "4aeb0af084892ad512165f0dd41980ecd8fd13dcf2faed45abc80dd057e18a53",
    "original_candidate_line_sha256": "4aeb0af084892ad512165f0dd41980ecd8fd13dcf2faed45abc80dd057e18a53"
  },
  "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-20261026-004-p6",
    "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": "5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2",
      "question_first_witness_sha256": "1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ca69d03adb8ed0d7f09a5e8d961497a8e1b8ef88cc7265255ddcd0a7f5f0cf09",
      "arithmetic_steps": 8
    },
    "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."
  }
}