{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r01-mathematical-induction-0058",
  "pilot_number": "R0009-05433",
  "number": 5433,
  "family_id": "qf-density12-induction-candidate-step-defect",
  "title": {
    "zh-Hant": "第 5433 題：候選遞推的基礎與步缺陷",
    "en": "Question 5433: Separate Base and Successor-Step Defects"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r01-mathematical-induction-0058",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-induction"
    ],
    "title_zh": "候選遞推的基礎與步缺陷",
    "stem_zh": "N從0起；有限歸納協議；目標初值s=1、步增量d=0；候選c(k)=-5k+(9)，n=1。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。",
    "answer": {
      "kind": "expression",
      "value": "(8, -5)",
      "explanation_zh": "候選基礎與步缺陷=(8, -5)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "候選遞推的基礎與步缺陷"
      ]
    },
    "variant": {
      "family_id": "qf-density12-induction-candidate-step-defect",
      "seed": 202610121058,
      "parameters": {
        "s": 1,
        "d": 0,
        "a": -5,
        "b": 9,
        "n": 1,
        "operation": "density12_induction_candidate_step_defect",
        "domain": "finite_induction_protocol",
        "semantic_template": "qf-density12-induction-candidate-step-defect/[\"8\",\"-5\"]"
      }
    },
    "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-20261012-r01-mathematical-induction-0058",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "000b924e212b1b8566bc5ba2bace9d58cba29c6721e54df060beff81b973379b",
      "family_id": "qf-density12-induction-candidate-step-defect",
      "seed": 202610121058,
      "semantic_template": "qf-density12-induction-candidate-step-defect/[\"8\",\"-5\"]"
    },
    "fields": {
      "title": "Separate Base and Successor-Step Defects",
      "stem": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=0; candidate c(k)=(-5)k+(9), n=1. Give signed integers (c(0)−s, c(n+1)−c(n)−d).",
      "answer": {
        "kind": "expression",
        "value": "(8, -5)",
        "explanation": "Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (8, -5)."
      },
      "skills": [
        "Separate Base and Successor-Step Defects"
      ],
      "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從0起；有限歸納協議；目標初值s=1、步增量d=0；候選c(k)=-5k+(9)，n=1。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。",
        "en": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=0; candidate c(k)=(-5)k+(9), n=1. Give signed integers (c(0)−s, c(n+1)−c(n)−d)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "natural indices start at0;finite induction protocol",
      "after": "index bookkeeping and finite values are not proofs of all P(n)",
      "relation": "implies",
      "rule_id": "finite_induction_claim_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "自然數索引不要求序列值非負；有限範圍或後繼索引的核對不自動提供命題真值或所有自然數的歸納證明。",
        "en": "Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals."
      },
      "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": "successor target index := sum(1, 1)",
      "after": "successor target index = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從當前非負索引加1得到後繼目標。",
        "en": "Add1 to the current natural index to obtain the successor target."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 successor target index，不以小數近似替代。",
          "en": "Recompute successor target index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "successor target index"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "candidate c(0) product := mul(-5, 0)",
      "after": "candidate c(0) product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將指定索引代入候選仿射式的乘法項。",
        "en": "Substitute the specified index into the candidate affine product term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(0) product，不以小數近似替代。",
          "en": "Recompute candidate c(0) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "0",
        "label": "candidate c(0) product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate c(0) value := sum(0, 9)",
      "after": "candidate c(0) value = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加候選常數b。",
        "en": "Then add the candidate constant b."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(0) value，不以小數近似替代。",
          "en": "Recompute candidate c(0) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "9"
        ],
        "output": "9",
        "label": "candidate c(0) value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate c(1) product := mul(-5, 1)",
      "after": "candidate c(1) product = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將指定索引代入候選仿射式的乘法項。",
        "en": "Substitute the specified index into the candidate affine product term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(1) product，不以小數近似替代。",
          "en": "Recompute candidate c(1) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-5",
        "label": "candidate c(1) product"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate c(1) value := sum(-5, 9)",
      "after": "candidate c(1) value = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加候選常數b。",
        "en": "Then add the candidate constant b."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(1) value，不以小數近似替代。",
          "en": "Recompute candidate c(1) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "9"
        ],
        "output": "4",
        "label": "candidate c(1) value"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "candidate c(2) product := mul(-5, 2)",
      "after": "candidate c(2) product = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將指定索引代入候選仿射式的乘法項。",
        "en": "Substitute the specified index into the candidate affine product term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(2) product，不以小數近似替代。",
          "en": "Recompute candidate c(2) product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "2"
        ],
        "output": "-10",
        "label": "candidate c(2) product"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate c(2) value := sum(-10, 9)",
      "after": "candidate c(2) value = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加候選常數b。",
        "en": "Then add the candidate constant b."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c(2) value，不以小數近似替代。",
          "en": "Recompute candidate c(2) value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "9"
        ],
        "output": "-1",
        "label": "candidate c(2) value"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "candidate base defect := sub(9, 1)",
      "after": "candidate base defect = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基礎缺陷為c(0)−s，不與步缺陷混用。",
        "en": "The base defect is c(0)−s,distinct from the step defect."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate base defect，不以小數近似替代。",
          "en": "Recompute candidate base defect with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "1"
        ],
        "output": "8",
        "label": "candidate base defect"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "candidate adjacent difference := sub(-1, 4)",
      "after": "candidate adjacent difference = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取相鄰候選值c(n+1)−c(n)。",
        "en": "First subtract adjacent candidate values c(n+1)−c(n)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate adjacent difference，不以小數近似替代。",
          "en": "Recompute candidate adjacent difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "4"
        ],
        "output": "-5",
        "label": "candidate adjacent difference"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "candidate step defect := sub(-5, 0)",
      "after": "candidate step defect = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再扣要求的增量d，保留有號缺陷。",
        "en": "Then subtract the required increment d,retaining the signed defect."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate step defect，不以小數近似替代。",
          "en": "Recompute candidate step defect with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "-5",
        "label": "candidate step defect"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "c(k)=-5k+9;specified n=1",
      "after": "base defect=8;step defect=-5",
      "relation": "implies",
      "rule_id": "affine_candidate_base_step_separation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題回報指定代入；若另做符號恆等化簡，可得基礎b−s及步a−d，這與只從有限測例外推一般命題不同。",
        "en": "Report the specified substitutions.A separate symbolic identity gives base b−s and step a−d for this affine model;that differs from extrapolating a general proposition from finitely many samples."
      },
      "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": [
      "8",
      "-5"
    ],
    "candidate_values": [
      "9",
      "4",
      "-1"
    ]
  },
  "conclusion": {
    "canonical_value": "(8, -5)",
    "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-20261012-r01-mathematical-induction-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r01-mathematical-induction-foundations/questions.jsonl",
      "batch_sha256": "4ff961382c9b0e9265156817fa51ccd4f7623e1942ae117f50b21701e1edb9f1",
      "original_line": 58,
      "original_line_sha256": "77bd57d798a4dbf39138182c35099de7da5b53296ba229be759e3d155faf9409",
      "question_compact_sha256": "000b924e212b1b8566bc5ba2bace9d58cba29c6721e54df060beff81b973379b",
      "english_row_compact_sha256": "6e8b1d3e209e8a6ee4111fb51fc8b3bd30e663448bd406cf10b4e76bebe46de0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r01-mathematical-induction-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r01-mathematical-induction-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-mathematical-induction": {
        "sha256": "53fe3a341f5fa25780e7aa742a25087665b257b3c91a3e4f718b50f368a8dd67",
        "english_sha256": "18d956a6519adafa325420611bdba90cada15d1a1db00caffeca5e4b4979cada",
        "original_path": "public/data/mko/mathematical-induction.json",
        "english_path": "public/data/i18n/editions/mko-mathematical-induction.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "57d2770c51a186a408b374e627ae4903b718639db16da9f5b6e8831f70297d8f",
    "original_candidate_line_sha256": "57d2770c51a186a408b374e627ae4903b718639db16da9f5b6e8831f70297d8f"
  },
  "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-20261026-005-p10",
    "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56",
      "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "987901dd572a8add7d93bd533d5b11445543c14a3486ad50dfa79aff1e49e375",
      "arithmetic_steps": 10
    },
    "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."
  }
}