{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-coordinate-plane",
  "semantic_family_id": "sf-coordinate-plane",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "座標平面"
  },
  "summary": {
    "zh-Hant": "固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。"
  },
  "statement": {
    "zh-Hant": "在指定平面直角座標系中，點的座標是有序對 (x,y)。x 軸滿足 y=0，y 軸滿足 x=0；兩者交於 (0,0)。軸外四象限由 x、y 的符號決定。"
  },
  "formula": {
    "tex": "P(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7a2515b2af601e5713a0311388890a869f9a5888a02615c21665028dd77c01be"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "固定平面上的原點、互相垂直的 x／y 有向軸與各軸單位。"
    },
    {
      "id": "a2",
      "zh-Hant": "座標為實數有序對，兩點比較使用同一座標系；本輪不推論距離或图形尺寸。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "point_coordinate_label",
      "role_zh": "點的代表記號",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "horizontal_coordinate",
      "role_zh": "第一座標",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "vertical_coordinate",
      "role_zh": "第二座標",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "兩條座標軸以實數值表示。"
    },
    {
      "id": "mko-point-line-plane",
      "title_zh": "點、線、面",
      "reason_zh": "平面上的點與兩條座標軸提供幾何語境。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "第一座標沿 x 軸方向讀取，第二座標沿 y 軸方向讀取。交換座標通常改變點。",
      "x、y 都非零時，(+,+)、(-,+)、(-,-)、(+,-) 分別為第一、二、三、四象限；軸上的點不屬於任何象限。原點同時在兩軸。",
      "在同一座標系中，兩點相等當且僅當兩個對應座標分別相等。座標值是明示資料，不能由示意圖比例猜測。"
    ],
    "common_misconception_zh": "把軸上或原點誤列為象限，或把有序對誤當可交換的無序集合。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示整数座標的角色與符號可重算；歐幾里得距離公式、座標變換和一般幾何定理另有義務。",
    "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": "綁定實數有序對、指定座標系與軸／象限的精確判定；距離另開語義橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-coordinate-plane.md 整理；2026-09-30 脈衝 R07，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-30"
  }
}