{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r01-natural-number-0030",
  "pilot_number": "R0008-04830",
  "number": 4830,
  "family_id": "qf-density11-natural-predecessor-pair",
  "title": {
    "zh-Hant": "第 4830 題：部分自然前驅的有序對",
    "en": "Question 4830: Partial Natural Predecessors"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r01-natural-number-0030",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-natural-number"
    ],
    "title_zh": "部分自然前驅的有序對",
    "stem_zh": "N包含0；有序值=[4, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。",
    "answer": {
      "kind": "expression",
      "value": "(3, 3)",
      "explanation_zh": "部分自然前驅=(3, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "部分自然前驅的有序對"
      ]
    },
    "variant": {
      "family_id": "qf-density11-natural-predecessor-pair",
      "seed": 202610111030,
      "parameters": {
        "values": [
          4,
          4
        ],
        "operation": "density11_natural_predecessor_pair",
        "domain": "N_including_zero",
        "semantic_template": "qf-density11-natural-predecessor-pair/[4,4]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261011-r01-natural-number-0030",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c0ec384f201d000c829b67ed7cc3ef42f0e992cd7c77204658e2ff565fa849e4",
      "family_id": "qf-density11-natural-predecessor-pair",
      "seed": 202610111030,
      "semantic_template": "qf-density11-natural-predecessor-pair/[4,4]"
    },
    "fields": {
      "title": "Partial Natural Predecessors",
      "stem": "N includes 0. The ordered values are [4, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.",
      "answer": {
        "kind": "expression",
        "value": "(3, 3)",
        "explanation": "Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (3, 3)."
      },
      "skills": [
        "Partial Natural Predecessors"
      ],
      "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": "N包含0；有序值=[4, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。",
        "en": "N includes 0. The ordered values are [4, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "N={0,1,2,...}",
      "after": "0 belongs;negative and nonintegral rational values do not",
      "relation": "implies",
      "rule_id": "natural_zero_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題自然數包含0；與有理數值、部分前驅及閉區間的規則分開判定。",
        "en": "This convention includes0 in N;distinguish membership,rational values,partial predecessors and closed intervals."
      },
      "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": "natural predecessor0 := sub(4, 1)",
      "after": "natural predecessor0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正自然數減1仍在N中。",
        "en": "Subtracting1 from a positive natural remains in N."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 natural predecessor0，不以小數近似替代。",
          "en": "Recompute natural predecessor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "1"
        ],
        "output": "3",
        "label": "natural predecessor0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "n=4",
      "after": "natural predecessor=3",
      "relation": "implies",
      "rule_id": "partial_predecessor_zero_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0沒有自然前驅，因N內任意數加1都正；不是截斷前驅pred(0)=0，也不返回−1作自然數。",
        "en": "Zero has no natural predecessor because every natural plus1 is positive;this is neither truncated pred(0)=0 nor a return of−1 as a natural."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "natural predecessor1 := sub(4, 1)",
      "after": "natural predecessor1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正自然數減1仍在N中。",
        "en": "Subtracting1 from a positive natural remains in N."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 natural predecessor1，不以小數近似替代。",
          "en": "Recompute natural predecessor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "1"
        ],
        "output": "3",
        "label": "natural predecessor1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "n=4",
      "after": "natural predecessor=3",
      "relation": "implies",
      "rule_id": "partial_predecessor_zero_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0沒有自然前驅，因N內任意數加1都正；不是截斷前驅pred(0)=0，也不返回−1作自然數。",
        "en": "Zero has no natural predecessor because every natural plus1 is positive;this is neither truncated pred(0)=0 nor a return of−1 as a natural."
      },
      "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": [
      3,
      3
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 3)",
    "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-20261008-advance-20261011-r01-natural-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r01-natural-number-foundations/questions.jsonl",
      "batch_sha256": "ccaa1444bd0d6cb142b205d76ade261dd201940cf4971f1370270518e95618a0",
      "original_line": 30,
      "original_line_sha256": "2dcde3ae3ee414a8d274c31f95f88c7938b03ec4673d32508e025d4837f387b8",
      "question_compact_sha256": "c0ec384f201d000c829b67ed7cc3ef42f0e992cd7c77204658e2ff565fa849e4",
      "english_row_compact_sha256": "4d481e0b2d0ea105f0a064fda69435db3e714ca747f8dd45e9554dfd68f9a747",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r01-natural-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r01-natural-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-natural-number": {
        "sha256": "222b72502fb661f3372012f602fd2f1eeef754b9e2069498c6b24234778730c4",
        "english_sha256": "f7444a7bfebcfed594478ed5e7c0f36cb1985114de58b5b473beddd1961c0bbc",
        "original_path": "public/data/mko/natural-number.json",
        "english_path": "public/data/i18n/editions/mko-natural-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6dc34e24b0f43c50e49365ca1a1153175f8661d5290c9236df5527bb047157eb",
    "original_candidate_sha256": "d1493c3ab406f3654a041ea44c57ca418ed775db6a82197fe7d099434cfdb475",
    "original_candidate_line_sha256": "d1493c3ab406f3654a041ea44c57ca418ed775db6a82197fe7d099434cfdb475"
  },
  "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-p9",
    "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": "b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289",
      "question_first_witness_sha256": "caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "471f81060aad4601bf86d08ecf4676a4501373221b6aa3a702019942cbef8368",
      "arithmetic_steps": 2
    },
    "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."
  }
}