{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r05-linear-function-0076",
  "pilot_number": "R0001-01176",
  "number": 1176,
  "family_id": "qf-affine-pointwise-comparison",
  "title": {
    "zh-Hant": "第 1176 題：仿射函數的單點比較",
    "en": "Question 1176: Affine Functions at One Input"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r05-linear-function-0076",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-function"
    ],
    "title_zh": "仿射函數的單點比較",
    "stem_zh": "在實數論域 R，f(x)=(1)x+(2)，g(x)=(1)x+(2)。只問 f(0) 與 g(0) 是否相等？",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "f(0)=2, g(0)=2。答案是是；單點輸出相同不代表全域相等。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "仿射函數的單點比較"
      ]
    },
    "variant": {
      "family_id": "qf-affine-pointwise-comparison",
      "seed": 202610035076,
      "parameters": {
        "m": 1,
        "b": 2,
        "u": 1,
        "v": 2,
        "x": 0,
        "domain": "reals",
        "operation": "compare_at_input",
        "semantic_template": "qf-affine-pointwise-comparison/{\"b\":2,\"domain\":\"reals\",\"m\":1,\"operation\":\"compare_at_input\",\"u\":1,\"v\":2,\"x\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r05-linear-function-0076",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "376948f2492dd57a7c67e5c6be33d46ea622cd4ac1d7227c488a0784b75f6bb2",
      "family_id": "qf-affine-pointwise-comparison",
      "seed": 202610035076,
      "semantic_template": "qf-affine-pointwise-comparison/{\"b\":2,\"domain\":\"reals\",\"m\":1,\"operation\":\"compare_at_input\",\"u\":1,\"v\":2,\"x\":0}"
    },
    "fields": {
      "title": "Affine Functions at One Input",
      "stem": "Over R, f(x)=(1)x+(2) and g(x)=(1)x+(2). At the single input x=0, are f(x) and g(x) equal?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "f(0)=2 and g(0)=2, so the answer is Yes. Equality at this input does not establish equality on the whole domain."
      },
      "skills": [
        "Affine Functions at One Input"
      ],
      "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": "在實數論域 R，f(x)=(1)x+(2)，g(x)=(1)x+(2)。只問 f(0) 與 g(0) 是否相等？",
        "en": "Over R, f(x)=(1)x+(2) and g(x)=(1)x+(2). At the single input x=0, are f(x) and g(x) equal?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f:R→R; f(x)=mx+b",
      "after": "m=0 gives a constant affine function",
      "relation": "implies",
      "rule_id": "affine_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "仿射函數允許m=0；不能把一次方程式a≠0的限制誤套到所有仿射函數。",
        "en": "An affine function allows m=0; do not impose the nondegenerate-equation condition on every affine function."
      },
      "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": "m·x := mul(1, 0)",
      "after": "m·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整代入同一輸入值，先做乘法。",
        "en": "Substitute the same input and multiply first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 m·x，不以小數近似替代。",
          "en": "Recompute m·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "m·x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "f(x) := sum(0, 2)",
      "after": "f(x) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加上常數截距。",
        "en": "Then add the constant intercept."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x)，不以小數近似替代。",
          "en": "Recompute f(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "f(x)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "u·x := mul(1, 0)",
      "after": "u·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二函數也用同一輸入，不能改成另一點。",
        "en": "Evaluate the second function at the same input, not a different point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 u·x，不以小數近似替代。",
          "en": "Recompute u·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "u·x"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "g(x) := sum(0, 2)",
      "after": "g(x) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完成第二函數的完整代值。",
        "en": "Complete the second function's substitution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x)，不以小數近似替代。",
          "en": "Recompute g(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "g(x)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "f(0)=2; g(0)=2",
      "after": "equal at this point: 是",
      "relation": "implies",
      "rule_id": "pointwise_not_global",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只判斷這一點是否相等；即使相等，也不代表全域相等。",
        "en": "Decide equality at this point only; agreement here does not imply global equality."
      },
      "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": {
    "f_value": "2",
    "g_value": "2",
    "equal": true
  },
  "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-20261003-r05-linear-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r05-linear-function-foundations/questions.jsonl",
      "batch_sha256": "ff5c60775dad16ef6f718760b27128b102226c42313362bb6c8a6a5a19addf7e",
      "original_line": 76,
      "original_line_sha256": "3d0bcc2a7a6fdb11baae49d4f419a8275df3fb083792c294f30e7c784e965b56",
      "question_compact_sha256": "376948f2492dd57a7c67e5c6be33d46ea622cd4ac1d7227c488a0784b75f6bb2",
      "english_row_compact_sha256": "fbc717627adb7ff379a263e0e56d4a22719bbadea5e495bd33bd1f0278783bb4",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r05-linear-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r05-linear-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-function": {
        "sha256": "718236b5b7dfecd9f4d276590f1ddc012cfd5df96864540a3840a2b4e6650abe",
        "english_sha256": "15d0d8ac9fe6a30744f0bbe21d2d94a8273271f0e4c8c1d4ed982b99f72e098b",
        "original_path": "public/data/mko/linear-function.json",
        "english_path": "public/data/i18n/editions/mko-linear-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b9fe70a7a264c90c20349be0449b2bd1a1b4d0e0a97092bc9fbcb7ca36ce0694",
    "original_candidate_sha256": "cf7f3060a6917a02ab06dd4760f723786781ee315c302053b1509f981b5d080a",
    "original_candidate_line_sha256": "cf7f3060a6917a02ab06dd4760f723786781ee315c302053b1509f981b5d080a"
  },
  "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-p2",
    "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": "c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc",
      "question_first_witness_sha256": "3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "69b9539229b80b22110afadd10cccd081303c3d7a0c131325c51936af0ce108b",
      "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."
  }
}