{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-orbit-fixed-periodic",
  "semantic_family_id": "sf-orbit-fixed-periodic",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "軌道、固定點與週期點"
  },
  "summary": {
    "zh-Hant": "反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。"
  },
  "statement": {
    "zh-Hant": "對自映射 T:X→X，x 的前向軌道為 x,T(x),T²(x),…。若 T(x)=x，x 是固定點；若存在最小正整數 p 使 T^p(x)=x，則 x 是最小週期 p 的週期點。最終進入循環不代表起點本身為週期點。"
  },
  "formula": {
    "tex": "T(x)=x",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "x",
        "glyph": "x"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f08bcfc604f3ee12b6c335f3ee962e877ab449d1b7c446489d4294fe61609dc5"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫採完整有限剩餘類狀態及仿射自映射，可枚舉至首次重複。"
    },
    {
      "id": "a2",
      "zh-Hant": "分類只適用指定有限模型；不由任意有限前綴推論一般無限系統非週期。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "state_update",
      "role_zh": "狀態更新自映射",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "initial_state",
      "role_zh": "指定起點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-discrete-dynamical-system",
      "title_zh": "離散時間動力系統",
      "reason_zh": "軌道由既定自映射反覆迭代。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "軌道按自然數時間排列。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "固定與週期條件是迭代等式。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "固定點是最小週期1的特殊情形；T^p(x)=x只證p是返回時間，仍要排除更小正返回時間。",
      "有限自映射從任一點出發終會重複，但若首次循環在起點之後才開始，起點只是前週期點。",
      "完整有限表可精確列出各點最小週期；這不外推連續系統或參數族的週期結構。"
    ],
    "common_misconception_zh": "把任何返回時間都叫最小週期，或把最後進入循環的起點也叫週期點。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限模型可分類指定點；有限前綴本身不證一般非週期性。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入前週期、吸引週期、穩定性與參數分岔。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-orbit-fixed-periodic.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-21 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}