{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261020-r07-compound-interest-0079",
  "pilot_number": "R0002-01829",
  "number": 1829,
  "family_id": "qf-app20-interest-interest-earned",
  "title": {
    "zh-Hant": "第 1829 題：固定期複利模型",
    "en": "Question 1829: Net Compound Interest Can Be Negative"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r07-compound-interest-0079",
    "batch_id": "batch-20261013-advance-20261020-r07-compound-interest-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-compound-interest"
    ],
    "title_zh": "固定期複利模型",
    "stem_zh": "固定每期利率且無現金流。操作=interest-interest-earned；資料={\"principal\":[115,4],\"rate\":[1,20],\"periods\":0}。請依 [principal,rate,periods,future,interest] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(115/4, 1/20, 0, 115/4, 0)",
      "explanation_zh": "應用基礎核對=(115/4, 1/20, 0, 115/4, 0)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；期別與利率慣例已固定；不是投資建議或市場預測。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "固定期複利模型"
      ]
    },
    "variant": {
      "family_id": "qf-app20-interest-interest-earned",
      "seed": 202610207079,
      "parameters": {
        "principal": [
          115,
          4
        ],
        "rate": [
          1,
          20
        ],
        "periods": 0,
        "domain": "Exact_fixed_period_compound_model",
        "operation": "interest-interest-earned",
        "semantic_template": "qf-app20-interest-interest-earned/{\"periods\":0,\"principal\":[115,4],\"rate\":[1,20]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261020-r07-compound-interest-0079",
    "batch_id": "batch-20261013-advance-20261020-r07-compound-interest-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9c8cbf924ced0ea87977486ecf0c0e3d9bf7cc3ae3fb966acc7e729ce035c71c",
      "family_id": "qf-app20-interest-interest-earned",
      "seed": 202610207079,
      "semantic_template": "qf-app20-interest-interest-earned/{\"periods\":0,\"principal\":[115,4],\"rate\":[1,20]}"
    },
    "fields": {
      "title": "Net Compound Interest Can Be Negative",
      "stem": "Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. principal=115/4, rate=1/20, periods=0. Give (principal, rate, periods, future value, future value−principal). A permitted negative rate may produce negative net interest.",
      "answer": {
        "kind": "expression",
        "value": "(115/4, 1/20, 0, 115/4, 0)",
        "explanation": "Subtract principal without taking an absolute value or clipping the result to zero. This is not investment advice or a market forecast. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (115/4, 1/20, 0, 115/4, 0)."
      },
      "skills": [
        "Net Compound Interest Can Be Negative"
      ],
      "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": "固定每期利率且無現金流。操作=interest-interest-earned；資料={\"principal\":[115,4],\"rate\":[1,20],\"periods\":0}。請依 [principal,rate,periods,future,interest] 欄序回答。",
        "en": "Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. principal=115/4, rate=1/20, periods=0. Give (principal, rate, periods, future value, future value−principal). A permitted negative rate may produce negative net interest."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "accumulation factor := sum(1, 1/20)",
      "after": "accumulation factor = 21/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每期利率加1才是乘到餘額的累積因子。",
        "en": "Add1 to the per-period rate to obtain the multiplicative accumulation factor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 accumulation factor，不以小數近似替代。",
          "en": "Recompute accumulation factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1/20"
        ],
        "output": "21/20",
        "label": "accumulation factor"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "per-period rate=1/20; factor=21/20>0",
      "after": "fixed-period exact recurrence B_(k+1)=(1+r)B_k",
      "relation": "implies",
      "rule_id": "compound_model_assumptions",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只使用固定期、固定利率、無新增收支的精確模型；不當作實際金融商品建議，也不提前作分幣四捨五入。",
        "en": "Use only the exact fixed-period,fixed-rate model with no extra cash flows;this is not product advice and no currency rounding is introduced."
      },
      "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": "B₀=115/4; periods=0",
      "after": "0 periods leave the principal unchanged",
      "relation": "implies",
      "rule_id": "principal_and_period_count",
      "conditions": [],
      "explanation": {
        "zh-Hant": "期數0的餘額就是本金，不能誤加一期間利息。",
        "en": "At0 periods the balance is the principal;do not add a period of interest."
      },
      "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": "B₀=P; B_(k+1)=(1+r)B_k",
      "after": "B_n=P(1+r)^n by induction",
      "relation": "implies",
      "rule_id": "compound_power_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基底期數0成立；若k期公式成立，乘同一因子即可得k+1期。",
        "en": "The0-period base case holds,and multiplying by the same factor proves the next induction case."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "(21/20)^0",
      "after": "(1+r)^n = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "21/20",
          "0"
        ],
        "output": "1",
        "label": "(1+r)^n"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "future balance := mul(115/4, 1)",
      "after": "future balance = 115/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以冪公式代回，與逐期餘額互相核對。",
        "en": "Substitute into the power formula and cross-check against the periodwise balances."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 future balance，不以小數近似替代。",
          "en": "Recompute future balance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "115/4",
          "1"
        ],
        "output": "115/4",
        "label": "future balance"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "net interest := sub(115/4, 115/4)",
      "after": "net interest = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "利息淨額是期末餘額減本金；負利率時可為負，不截成0。",
        "en": "Net interest is final balance minus principal;it may be negative at a negative rate and must not be clipped to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 net interest，不以小數近似替代。",
          "en": "Recompute net interest with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "115/4",
          "115/4"
        ],
        "output": "0",
        "label": "net interest"
      }
    }
  ],
  "math_model": {
    "result": [
      "115/4",
      "1/20",
      0,
      "115/4",
      "0"
    ],
    "balances": [
      "115/4"
    ]
  },
  "conclusion": {
    "canonical_value": "(115/4, 1/20, 0, 115/4, 0)",
    "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": "d7c8f00668665182d187079e70c7ecd9a58c5530991543150c6071ec31df1010"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "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-p3",
    "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": {
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    "bilingual_semantics": true,
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    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}