{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r05-linear-function-0058",
  "pilot_number": "R0001-01158",
  "number": 1158,
  "family_id": "qf-affine-zero-set",
  "title": {
    "zh-Hant": "第 1158 題：仿射函數的完整零點集合",
    "en": "Question 1158: Complete Zero Set of an Affine Function"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r05-linear-function-0058",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-function"
    ],
    "title_zh": "仿射函數的完整零點集合",
    "stem_zh": "在實數論域 R，f(x)=(-5)x+(0)。求全部零點，以 R、∅ 或最簡有理數集合回答。",
    "answer": {
      "kind": "expression",
      "value": "{0/1}",
      "explanation_zh": "零點集={0/1}。m≠0 時零點唯一；m=0 且 b=0 時全部實數皆為零點，m=0 且 b≠0 時無零點。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "仿射函數的完整零點集合"
      ]
    },
    "variant": {
      "family_id": "qf-affine-zero-set",
      "seed": 202610035058,
      "parameters": {
        "m": -5,
        "b": 0,
        "domain": "reals",
        "operation": "zero_set",
        "semantic_template": "qf-affine-zero-set/{\"b\":0,\"domain\":\"reals\",\"m\":-5,\"operation\":\"zero_set\"}"
      }
    },
    "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-0058",
    "batch_id": "batch-20261003-r05-linear-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4c6c4aaaa96592303920d776e9898e057163948ac7c46644b5a5b55aa955c858",
      "family_id": "qf-affine-zero-set",
      "seed": 202610035058,
      "semantic_template": "qf-affine-zero-set/{\"b\":0,\"domain\":\"reals\",\"m\":-5,\"operation\":\"zero_set\"}"
    },
    "fields": {
      "title": "Complete Zero Set of an Affine Function",
      "stem": "Over R, f(x)=(-5)x+(0). Find its complete zero set, using R, ∅, or a set of reduced rational numbers.",
      "answer": {
        "kind": "expression",
        "value": "{0/1}",
        "explanation": "The zero set is {0/1}. For m≠0, −b/m is the unique zero; for m=0 and b=0 every real input is a zero, while for m=0 and b≠0 there are none."
      },
      "skills": [
        "Complete Zero Set of an Affine Function"
      ],
      "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)=(-5)x+(0)。求全部零點，以 R、∅ 或最簡有理數集合回答。",
        "en": "Over R, f(x)=(-5)x+(0). Find its complete zero set, using R, ∅, or a set of reduced rational numbers."
      }
    }
  ],
  "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": "-5x+(0)=0",
      "after": "m≠0: 是",
      "relation": "implies",
      "rule_id": "zero_set_case",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先判斷是否可除以m，常數函數需獨立分類。",
        "en": "Check whether division by m is allowed; classify constant functions separately."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "root := div(0, -5)",
      "after": "root = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "移項後同除非零m，得到唯一候選。",
        "en": "Move the constant and divide by nonzero m to obtain the unique candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 root，不以小數近似替代。",
          "en": "Recompute root with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "-5"
        ],
        "output": "0",
        "label": "root"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "m·root := mul(-5, 0)",
      "after": "m·root = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將候選代回一次項。",
        "en": "Substitute into the linear term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 m·root，不以小數近似替代。",
          "en": "Recompute m·root with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "0",
        "label": "m·root"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "f(root) := sum(0, 0)",
      "after": "f(root) = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "核對函數值為0。",
        "en": "Verify that the function value is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(root)，不以小數近似替代。",
          "en": "Recompute f(root) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "f(root)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "m(x₁−x₂)=0 with m≠0",
      "after": "x₁=x₂",
      "relation": "implies",
      "rule_id": "unique_affine_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個零點若存在，相減並約去非零m，必相等。",
        "en": "Any two zeros must coincide after subtraction and cancellation of nonzero m."
      },
      "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": {
    "branch": "unique",
    "root": "0"
  },
  "conclusion": {
    "canonical_value": "{0/1}",
    "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."
  },
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    "original_candidate_line_sha256": "b2794a5c03439d1c49a862776d5d414903c871cc4aaee120b06dc5570f644725"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "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": {
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    "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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}