{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-recurrence-relation-01",
  "mko_id": "mko-recurrence-relation",
  "title": {
    "zh-Hant": "oldest-to-newest二階遞迴trace",
    "en": "An oldest-to-newest order-two recurrence trace"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-recurrence-relation-01",
    "source": {
      "mko_id": "mko-recurrence-relation",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "972d0f75cb7afb4eadac89d2faf5c5f9f37f9f1c30ba6672dcf403faf3747056",
      "english_sha256": "9193c49f8b1b765f25b8e7b38967ac6f4793fe157214358b70a3498487a137a1"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "對齊state、係數與全部初值，生成a5並證唯一性責任。",
      "en": "Align states, coefficients and all initial values, generate a5 and explain uniqueness."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "order r=2，initial_values=(a0,a1)=(1,3)，coefficients=(2,1)，constant=1；所有n≥2用a_n=1+2a_(n-2)+a_(n-1)。",
          "en": "Order r=2, initial_values=(a0,a1)=(1,3), coefficients=(2,1), constant=1; for all n≥2 use a_n=1+2a_(n-2)+a_(n-1)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "A_1=(a_0,a_1)=(1,3)",
        "after_tex": "F(u,v)=(v,1+2u+v)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "最舊u乘2、最新v乘1，再左移append。",
          "en": "Multiply the oldest u by 2 and newest v by 1, then shift left and append."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能倒轉係數為u+2v。",
            "en": "Do not reverse coefficients to u+2v."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "a_2=1+2(1)+3",
        "after_tex": "a_2=6,\\quad A_2=(3,6)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一次update保留舊最新3作新state第一項。",
          "en": "The first update retains old newest3 as the first state entry."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "state不是只記新scalar6。",
            "en": "The state is not merely the new scalar6."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "a_3=1+2(3)+6",
        "after_tex": "a_3=13,\\quad A_3=(6,13)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "從A2正確對齊求下一項。",
          "en": "Align A2 correctly to compute the next term."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "係數2乘3而非6。",
            "en": "The coefficient 2 multiplies3, not 6."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "a_4=1+2(6)+13",
        "after_tex": "a_4=26,\\quad A_4=(13,26)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "再次shift後append26。",
          "en": "Shift again and append26."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "constant1在每次update都存在。",
            "en": "The constant 1 appears at every update."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "a_5=1+2(13)+26",
        "after_tex": "a_5=53,\\quad A_5=(26,53)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限target index5已完整生成。",
          "en": "The finite target index 5 has been fully generated."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "每個中間值為明示安全整數。",
            "en": "Every intermediate value is an explicitly small integer."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "a_0=b_0,\\ a_1=b_1\\quad\\text{and same recurrence}",
        "after_tex": "(a_{n-2},a_{n-1})=(b_{n-2},b_{n-1})\\implies a_n=b_n",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一single-valued update把相同前兩項送到相同下一項。",
          "en": "The same single-valued update sends equal preceding pairs to equal next terms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "歸納基礎需要兩個初值，不能只有一個。",
            "en": "The induction base requires both initial values."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "a_0=b_0=1,\\ a_1=b_1=3",
        "after_tex": "a_n=b_n\\quad\\text{for every }n\\ge0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以兩初值為基礎、上一項推導為step，數學歸納給唯一序列。",
          "en": "Using the two initial values as base and the preceding implication as step, induction gives sequence uniqueness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "唯一性不等於已證closed form或growth rate。",
            "en": "Uniqueness does not prove a closed form or growth rate."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "a_0=1\\text{ only},\\quad a_1=t",
        "after_tex": "a_2=3+t\\text{ depends on }t",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "缺第二初值時，不同t產生不同合法序列，不能稱唯一。",
          "en": "Without the second initial value, different t produce different valid sequences, so uniqueness fails."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "有限trace完成不宣稱收斂或漸近。",
            "en": "Completing a finite trace asserts no convergence or asymptotics."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "prefix為1、3、6、13、26、53；兩初值加total rule確定唯一序列。",
      "en": "The prefix is 1,3,6,13,26,53; two initial values and the total rule determine a unique sequence."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "a2=1+1+2乘3=8。",
          "en": "a2=1+1+2 times 3=8."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "係數順序最舊乘2，正確值6。",
          "en": "The oldest term gets coefficient 2, yielding6."
        }
      },
      {
        "claim": {
          "zh-Hant": "只給a0=1已唯一。",
          "en": "a0=1 alone gives uniqueness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "a1=t時a2=3+t仍自由。",
          "en": "a1=t leaves a2=3+t variable."
        }
      },
      {
        "claim": {
          "zh-Hant": "六項上升已證發散。",
          "en": "Six increasing terms prove divergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "有限prefix不能證無限tail性質。",
          "en": "A finite prefix cannot prove an infinite-tail property."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "r階遞迴須有全部r個初值，係數由最舊到最新對齊；狀態左移後附加新項。有限前綴不證閉式、收斂或漸近性。",
          "en": "An order-r recurrence needs all r initial values and coefficients aligned oldest to newest. Shift the state left and append the new term. A finite prefix proves no closed form, convergence or asymptotics."
        },
        "source": {
          "mko_id": "mko-recurrence-relation",
          "sha256": "972d0f75cb7afb4eadac89d2faf5c5f9f37f9f1c30ba6672dcf403faf3747056",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "order-r state按最舊到最新排列；single-valued整數update為total；數學歸納可從全部初值推唯一性。",
          "en": "Order-r states are ordered oldest to newest; the single-valued integer update is total; induction from all initial values proves uniqueness."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1+2*1+3",
        "rhs": "6"
      },
      {
        "id": "N02",
        "lhs": "1+2*13+26",
        "rhs": "53"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限exact integer trace；不猜closed form。",
      "en": "A finite exact integer trace, without guessing a closed form."
    },
    "md_symbols": [
      [
        "A_n",
        "最近兩項state",
        "latest-two-term state"
      ],
      [
        "F",
        "shift-append update",
        "shift-and-append update"
      ]
    ],
    "md_error_controls": [
      {
        "title": "係數對齊 / Coefficient alignment",
        "zh": "(2,1)分別乘a_(n-2)、a_(n-1)。",
        "en": "Pair(2,1) with a_(n-2),a_(n-1)."
      },
      {
        "title": "state width / State width",
        "zh": "每次state保留兩項，先shift再append。",
        "en": "Keep two terms in every state, shifting before appending."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite.",
    "english_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
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}