{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r07-discrete-logistic-growth-0014",
  "pilot_number": "R0003-02389",
  "number": 2389,
  "family_id": "qf-app21-logistic-single-step",
  "title": {
    "zh-Hant": "第 2389 題：離散logistic增量",
    "en": "Question 2389: One Exact Capacity-Limited Discrete Increment"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r07-discrete-logistic-growth-0014",
    "batch_id": "batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-discrete-logistic-growth"
    ],
    "title_zh": "離散logistic增量",
    "stem_zh": "固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[13,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(13/5, 3/4, 8, 3133/800)",
      "explanation_zh": "應用擴展核對=(13/5, 3/4, 8, 3133/800)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "離散logistic增量"
      ]
    },
    "variant": {
      "family_id": "qf-app21-logistic-single-step",
      "seed": 202610217014,
      "parameters": {
        "current": [
          13,
          5
        ],
        "rate": [
          3,
          4
        ],
        "capacity": [
          8,
          1
        ],
        "domain": "Exact_discrete_logistic_increment_model",
        "operation": "logistic-single-step",
        "semantic_template": "qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[3,4]}"
      }
    },
    "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、真實學生診斷、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": {
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    "question_id": "q-20261013-advance-20261021-r07-discrete-logistic-growth-0014",
    "batch_id": "batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7da7227506c626552563068a095186609f4393dfafd9cae2630024b244465f8a",
      "family_id": "qf-app21-logistic-single-step",
      "seed": 202610217014,
      "semantic_template": "qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[3,4]}"
    },
    "fields": {
      "title": "One Exact Capacity-Limited Discrete Increment",
      "stem": "Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).",
      "answer": {
        "kind": "expression",
        "value": "(13/5, 3/4, 8, 3133/800)",
        "explanation": "Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13/5, 3/4, 8, 3133/800)."
      },
      "skills": [
        "One Exact Capacity-Limited Discrete Increment"
      ],
      "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": "固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[13,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。",
        "en": "Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "0≤r=3/4≤1; K=8>0; 0≤x=13/5≤K",
      "after": "T(x)=x+r·x·(1−x/K)",
      "relation": "implies",
      "rule_id": "logistic_increment_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是原值加增量的固定模型，不是另一種直接乘型r·x·(1−x/K)。",
        "en": "This fixed model adds the increment to the old state;it is not the different direct-product map r·x·(1−x/K)."
      },
      "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": "x/K at step1 := div(13/5, 8)",
      "after": "x/K at step1 = 13/40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "容量K為正，因此可求標準化狀態。",
        "en": "Positive K permits normalization of the state."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x/K at step1，不以小數近似替代。",
          "en": "Recompute x/K at step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "13/5",
          "8"
        ],
        "output": "13/40",
        "label": "x/K at step1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "1−x/K at step1 := sub(1, 13/40)",
      "after": "1−x/K at step1 = 27/40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此回饋因子在宣告區間中非負。",
        "en": "This feedback factor is nonnegative in the declared interval."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 1−x/K at step1，不以小數近似替代。",
          "en": "Recompute 1−x/K at step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "13/40"
        ],
        "output": "27/40",
        "label": "1−x/K at step1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "x(1−x/K) at step1 := mul(13/5, 27/40)",
      "after": "x(1−x/K) at step1 = 351/200",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘當前狀態與剩餘比例。",
        "en": "Multiply the current state by the remaining proportion first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x(1−x/K) at step1，不以小數近似替代。",
          "en": "Recompute x(1−x/K) at step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13/5",
          "27/40"
        ],
        "output": "351/200",
        "label": "x(1−x/K) at step1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "increment at step1 := mul(3/4, 351/200)",
      "after": "increment at step1 = 1053/800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再乘增量率r。",
        "en": "Then multiply by the increment rate r."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 increment at step1，不以小數近似替代。",
          "en": "Recompute increment at step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/4",
          "351/200"
        ],
        "output": "1053/800",
        "label": "increment at step1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "next state at step1 := sum(13/5, 1053/800)",
      "after": "next state at step1 = 3133/800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須把原本x加回，不能漏掉加號。",
        "en": "Add the original x back;the addition must not be omitted."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 next state at step1，不以小數近似替代。",
          "en": "Recompute next state at step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13/5",
          "1053/800"
        ],
        "output": "3133/800",
        "label": "next state at step1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "T(x)−x=r·x·(1−x/K)≥0; K−T(x)=(K−x)(1−r·x/K)≥0",
      "after": "0≤3133/800≤8",
      "relation": "implies",
      "rule_id": "qualified_interval_preservation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個因子不等式由0≤r≤1與0≤x≤K推出，證明此模型保持區間，不只靠幾個樣本點猜測。",
        "en": "The factor inequalities follow from0≤r≤1 and0≤x≤K,proving interval preservation for this model rather than guessing from 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": [
      "13/5",
      "3/4",
      "8",
      "3133/800"
    ]
  },
  "conclusion": {
    "canonical_value": "(13/5, 3/4, 8, 3133/800)",
    "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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      "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."
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  "revision": {
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    "review_date": "2026-10-26",
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    "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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}