{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-power-root",
  "semantic_family_id": "sf-power-root",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "自然數冪與主平方根"
  },
  "summary": {
    "zh-Hant": "本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。"
  },
  "statement": {
    "zh-Hant": "對實數 a 與自然數 n，採空乘積慣例 a^0=1（本代數語境也包括 0^0=1），並由 a^(n+1)=a^n·a 定義冪。對 r≥0，主平方根 √r 是唯一滿足 y≥0 且 y²=r 的實數 y。"
  },
  "formula": {
    "tex": "a^{n}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mi>a</mi><mi>n</mi></msup></math>",
    "semantic_ast": {
      "type": "power",
      "base": {
        "type": "symbol",
        "name": "a",
        "glyph": "a"
      },
      "exponent": {
        "type": "symbol",
        "name": "n",
        "glyph": "n"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9db51281790edeb4b912c12a5fb96a59aae95a4efbbb7da675b81111c49c3931"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "n∈N={0,1,2,…}；本輪不處理負指數、非整數指數或複數分支。"
    },
    {
      "id": "a2",
      "zh-Hant": "主平方根只在 r≥0 的實數論域定義；0^0=1 是此處的代數慣例，不是極限運算規則。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "real_base",
      "role_zh": "實數底數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "natural_exponent",
      "role_zh": "自然數指數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "自然數冪以乘法遞迴定義。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "主平方根以非負實數論域和大小關係選定分支。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "a^n 不是 a·n。n=0 時按空乘積得到 1；n>0 時是 n 個 a 的乘積。負底數在正奇數冪下為負、正偶數冪下為正；零底數在正冪下仍為零。",
      "√0=0；正 r 的主平方根取非負分支。方程 y²=r 的實數解集在 r>0 時有兩根 ±√r，在 r=0 時只有零，在 r<0 時為空。因此 √(a²)=|a|，不可無條件寫成 a。",
      "題庫採整數底數、非負整數指數和完全平方數的精確切片，另檢查負根號輸入的論域。未來 Mathlib Real.sqrt 的負輸入補零語義必須經 r≥0 的橋接前提；不能把補零當成此處的課本根值。"
    ],
    "common_misconception_zh": "把主平方根寫成 ± 值、把 √(a²) 當成 a，或把 0^0 的局部代數慣例套到極限。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪案例可精確重算；一般平方根存在唯一性及冪法則仍未附形式 Evidence。",
    "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": "綁定自然数冪的空乘積遞迴、主根非負前提、sqrt_sq_eq_abs 及 Real.sqrt 的補零邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-power-root.md 整理；2026-10-03 脈衝 R01，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}