{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-binomial-theorem",
  "semantic_family_id": "sf-binomial-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "二項式定理"
  },
  "summary": {
    "zh-Hant": "交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。"
  },
  "statement": {
    "zh-Hant": "n∈N_0，(x+y)^n=sum_{k=0}^n C(n,k)x^(n-k)y^k，coefficients exact且indices/exponents不可錯置。n=0給constant1。combinatorial witness選k個y factors；negative/noninteger exponent與noncommuting variables不在本輪。"
  },
  "formula": {
    "tex": "(x+y)^n",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mi>n</mi></msup></math>",
    "semantic_ast": {
      "type": "power",
      "base": {
        "type": "group",
        "body": {
          "type": "addition",
          "operands": [
            {
              "type": "symbol",
              "name": "x",
              "glyph": "x"
            },
            {
              "type": "symbol",
              "name": "y",
              "glyph": "y"
            }
          ]
        }
      },
      "exponent": {
        "type": "symbol",
        "name": "n",
        "glyph": "n"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0249dc6c2d970216953af860cd5f5cc54163852426af08edc95b37ce92456345"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "n為非負安全整數，k介於0與n。"
    },
    {
      "id": "a2",
      "zh-Hant": "evaluation companions限exact integers。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "first_commuting_term",
      "role_zh": "第一交換項",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "second_commuting_term",
      "role_zh": "第二交換項",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "natural_exponent",
      "role_zh": "非負整數指數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-permutation-combination",
      "title_zh": "排列與組合",
      "reason_zh": "binomial coefficients。"
    },
    {
      "id": "mko-mathematical-induction",
      "title_zh": "數學歸納法",
      "reason_zh": "一般n proof。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "commutative expansion。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "k從0到n全部包含。",
      "coefficient symmetry與factorial formula exact。",
      "n=0與domain boundary分開。"
    ],
    "common_misconception_zh": "交換k與n-k指數、漏端點或套negative exponent finite expansion。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "不無條件套用noncommutative variables。",
    "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Catalan與generating functions。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-binomial-theorem.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-09 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}