{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-circle",
  "semantic_family_id": "sf-circle",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "圓的邊界與座標判定"
  },
  "summary": {
    "zh-Hant": "圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。"
  },
  "statement": {
    "zh-Hant": "在歐氏平面固定中心 O 與半徑 r≥0，C(O,r)={P:d(P,O)=r}；r>0 是通常的非退化圓，r=0 是僅含 O 的退化點集。點距中心小於、等於、大於 r 分別判為內、圓上、外；直徑長為 2r。"
  },
  "formula": {
    "tex": "C(O,r)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>O</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "O",
          "glyph": "O"
        },
        {
          "type": "symbol",
          "name": "r",
          "glyph": "r"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ab607cdf9526f24c9c40b026fa546b2ff680a832015f18840838d0a11e638d77"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "座標題與距離條目同採等單位正交框架，整數中心、點及非負整數半徑。"
    },
    {
      "id": "a2",
      "zh-Hant": "負半徑不在本輪論域；r=0 沒有內部，中心在圓上。"
    },
    {
      "id": "a3",
      "zh-Hant": "軸向點 O±(r,0) 或 O±(0,r) 均在圓上；r=0 時四種表示是同一點。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "circle_boundary",
      "role_zh": "圓邊界點集",
      "scope": "representative_formula"
    },
    {
      "token": "O",
      "role": "center_point",
      "role_zh": "圓心",
      "scope": "representative_formula"
    },
    {
      "token": "r",
      "role": "nonnegative_radius",
      "role_zh": "非負半徑",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-euclidean-length",
      "title_zh": "歐氏長度",
      "reason_zh": "半徑及直徑使用長度單位。"
    },
    {
      "id": "mko-point-line-plane",
      "title_zh": "點、線、面",
      "reason_zh": "圓是平面中的點集。"
    },
    {
      "id": "mko-distance-formula",
      "title_zh": "等單位正交座標的歐氏距離",
      "reason_zh": "整數座標判定比較平方距離。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "對 r≥0，可把 d(P,O)=r 等價改成距離平方等於 r²；小於與大於同樣由非負平方比較。圓內的點不是圓邊界的成員。",
      "r=0 時只中心滿足等距條件；其他點都在外，直徑為零。本輪特別稱它退化點集，不混稱正半徑曲線。",
      "軸向座標點以明示方向精確構造，不依螢幕圖形目測。圓心平移對成員判定不構成新計算，直徑題也不靠無關圓心變化充量。"
    ],
    "common_misconception_zh": "把圓當圓盤，接受負半徑，或在零半徑判出內部點。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數半徑及座標判定可重算；一般圓、圓盤與球面 API 的形式橋仍待建立。",
    "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": "綁定 EuclideanSpace 上 Metric.sphere／ball，分開正半徑與零半徑。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-circle.md 整理；2026-10-05 脈衝 R05（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}