{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r05-recurrence-relation-0018",
  "pilot_number": "R0044-22593",
  "number": 22593,
  "family_id": "qf-mod41-recurrence-specification-qualification",
  "title": {
    "zh-Hant": "第 22593 題：遞迴關係",
    "en": "Question 22593: An Order-r Total Rule Needs All r Initial Values"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r05-recurrence-relation-0018",
    "batch_id": "batch-20261014-advance-20261110-r05-recurrence-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-recurrence-relation"
    ],
    "title_zh": "遞迴關係",
    "stem_zh": "initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], 是, 是, rule_applies_for_n_ge_4)",
      "explanation_zh": "Wave6第四層核對=(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], 是, 是, rule_applies_for_n_ge_4)；依完整明示資料獨立重算；state shift-left後append新scalar term；order-r recurrence須有r個必要初值才唯一決定forward prefix；有限生成prefix不證closed form、收斂或漸近性；缺初值或多值rule不得宣稱unique。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "遞迴關係"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-recurrence-specification-qualification",
      "seed": 202611105018,
      "parameters": {
        "order": 4,
        "coefficients": [
          2,
          4,
          6,
          8
        ],
        "constant": 0,
        "initial_values": [
          17,
          18,
          19,
          20
        ],
        "domain": "Exact_total_linear_recurrence",
        "operation": "recurrence-specification-qualification",
        "semantic_template": "qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r05-recurrence-relation-0018",
    "batch_id": "batch-20261014-advance-20261110-r05-recurrence-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2733198e89792bf65679eff985e862e0111f8a09648b59b1162a65fa36c69765",
      "family_id": "qf-mod41-recurrence-specification-qualification",
      "seed": 202611105018,
      "semantic_template": "qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4}"
    },
    "fields": {
      "title": "An Order-r Total Rule Needs All r Initial Values",
      "stem": "Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).",
      "answer": {
        "kind": "expression",
        "value": "(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4)",
        "explanation": "Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4)."
      },
      "skills": [
        "An Order-r Total Rule Needs All r Initial Values"
      ],
      "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": "initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-specification-qualification；資料={\"order\":4,\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=recurrence-specification-qualification;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。",
        "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness."
      },
      "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": "recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[2, 4, 6, 8];constant=0;ALL initial a_0..a_(r-1)=[17, 18, 19, 20]",
      "after": "for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued",
      "relation": "implies",
      "rule_id": "recurrence_complete_initial_state_and_total_single_value_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定r個初值與coefficient的oldest-to-newest角色；已知前r項時，每個下一項只有一個值。這是forward定義，不是closed form或收斂聲稱。",
        "en": "Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim."
      },
      "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": "recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(2, 17)",
      "after": "recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 34",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "17"
        ],
        "output": "34",
        "label": "recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 34)",
      "after": "recurrence ORIGINAL n=4 partial sum after coefficient0 = 34",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient0，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "34"
        ],
        "output": "34",
        "label": "recurrence ORIGINAL n=4 partial sum after coefficient0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 18)",
      "after": "recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 72",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "18"
        ],
        "output": "72",
        "label": "recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(34, 72)",
      "after": "recurrence ORIGINAL n=4 partial sum after coefficient1 = 106",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient1，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "34",
          "72"
        ],
        "output": "106",
        "label": "recurrence ORIGINAL n=4 partial sum after coefficient1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 19)",
      "after": "recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 114",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "19"
        ],
        "output": "114",
        "label": "recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(106, 114)",
      "after": "recurrence ORIGINAL n=4 partial sum after coefficient2 = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient2，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "106",
          "114"
        ],
        "output": "220",
        "label": "recurrence ORIGINAL n=4 partial sum after coefficient2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 20)",
      "after": "recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 160",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "20"
        ],
        "output": "160",
        "label": "recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(220, 160)",
      "after": "recurrence ORIGINAL n=4 partial sum after coefficient3 = 380",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。",
        "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient3，不以小數近似替代。",
          "en": "Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "160"
        ],
        "output": "380",
        "label": "recurrence ORIGINAL n=4 partial sum after coefficient3"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[17, 18, 19, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 17, 34], [1, 1, 4, 18, 72], [2, 2, 6, 19, 114], [3, 3, 8, 20, 160]]",
      "after": "new a_4=380;next state=[18, 19, 20, 380];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380]",
      "relation": "implies",
      "rule_id": "recurrence_EVERY_state_update_shift_left_append_exact",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個coef只乘指定原索引；逐項加constant並核來源安全整數範圍。state左移後append新scalar，不反轉係數或拿已append值回填同一更新。",
        "en": "Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "recurrence ORIGINAL:complete ORIGINAL initial state=[17, 18, 19, 20];actual generated prefix=[17, 18, 19, 20, 380]",
      "after": "ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only",
      "relation": "implies",
      "rule_id": "recurrence_forward_uniqueness_induction_not_analytic_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "歸納先有全部r個初值相等，再由同一單值規則得到下一項相等；重複即可保整個forward prefix。有限生成不證closed form、漸近或解析性。",
        "en": "Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity."
      },
      "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": [
      4,
      [
        2,
        4,
        6,
        8
      ],
      0,
      [
        17,
        18,
        19,
        20
      ],
      true,
      true,
      "rule_applies_for_n_ge_4"
    ]
  },
  "conclusion": {
    "canonical_value": "(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], 是, 是, rule_applies_for_n_ge_4)",
    "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",
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      "batch_id": "batch-20261014-advance-20261110-r05-recurrence-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r05-recurrence-relation-foundations/questions.jsonl",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r05-recurrence-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r05-recurrence-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "english_path": "public/data/i18n/editions/mko-recurrence-relation.en.json"
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    "original_candidate_line_sha256": "a0d473dc097bd55c0383eef7187cc59b135b5021444185b0a0734e31c352728a",
    "mko_binding_revision": {
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      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
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      "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."
  },
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    "id": "review-20261026-009-p45",
    "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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    "bilingual_semantics": true,
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}