{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-similarity",
  "semantic_family_id": "sf-similarity",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "正倍率的標記三角形相似切片"
  },
  "summary": {
    "zh-Hant": "指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。"
  },
  "statement": {
    "zh-Hant": "對明示 Ai↔Bi 的兩個非退化歐氏平面三角形，本輪正倍率 k 的 SSS 相似切片要求每組對應邊長 B_iB_j=k A_iA_j。等價地三組平方長度都乘 k²；若給定A的正面積值，B的面積為 k² 倍。鏡射不改變正長度倍率。"
  },
  "formula": {
    "tex": "S(A,B)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>S</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "S",
        "glyph": "S"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6612f0061bbaf6750ae8da0b12959ca9ca476e0fa1c3fe5faa540e3f695baccc"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "兩個三角形非退化、同一等單位正交框架，對應與倍率方向均明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫 k=p/q，q>0，除倍率資格題外 p>0；資格題允許p≤0並答否。"
    },
    {
      "id": "a3",
      "zh-Hant": "指定倍率不符合時，只否定這個倍率見證，不代表不存在其他相似倍率。面積題已給正有理面積，不聲稱矩形面積物件已證成一般三角形面積理論。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "positive_scale_labeled_similarity",
      "role_zh": "指定正倍率的標記相似關係",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "source_triangle",
      "role_zh": "倍率來源三角形",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "target_triangle",
      "role_zh": "倍率目標三角形",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-congruence",
      "title_zh": "明示對應的非退化三角形全等",
      "reason_zh": "非退化及明示對應的距離判定。"
    },
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "正有理倍率與比例方向。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "自然數冪與主平方根",
      "reason_zh": "平方長度及倍率平方。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "長度的倍率是k，平方長度的倍率是k²。非退化下對應邊長為正，可以以整數交叉乘積 B²q²=A²p² 精確檢查。",
      "面積題是給定相似三角形、正倍率與已知正面積的條件計算，不以座標圖形猜面積，也不外推為一般測度理論。",
      "對座標做係數−2的均勻縮放仍可得到正長度倍率2；負座標係數與負長度倍率不是同一件事。倍率零會退化，不屬本條目的非退化相似。"
    ],
    "common_misconception_zh": "長度和面積都只乘k、把平方倍率當長度倍率，或以錯誤的指定k否定所有相似可能。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "正有理倍率切片可精確重算；一般相似映射、SSS及三角形面積比的形式 Evidence 未建立。",
    "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": "依固定版綁定正比例對應距離與 k² 面積比；不借用矩形／格子模型的未有形式證據。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-similarity.md 整理；2026-10-06 脈衝 R06（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}