{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-mobius-function",
  "semantic_family_id": "sf-mobius-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Möbius函數"
  },
  "summary": {
    "zh-Hant": "Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。"
  },
  "statement": {
    "zh-Hant": "完整UPF certificate提供multiplicities與distinct-prime count。factor order不影響。sum_{d|n}mu(d)=1 if n=1 else0作精確cross-check。0與negative integers不在domain；incomplete trial division不得分類。"
  },
  "formula": {
    "tex": "mu(n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>mu</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "mu",
        "glyph": "mu"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cb9e81b7b72c81cf53ec49f4583888efa9a876eb30634064ca9f25fac40d4d59"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "domain為positive integer且factorization完整。"
    },
    {
      "id": "a2",
      "zh-Hant": "bounded companions完整trial divide至sqrt remaining。"
    }
  ],
  "symbols": [
    {
      "token": "mu",
      "role": "mobius_function",
      "role_zh": "Möbius函數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "positive_integer",
      "role_zh": "正整數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-prime-number",
      "title_zh": "自然數質數",
      "reason_zh": "prime factors。"
    },
    {
      "id": "mko-unique-prime-factorization",
      "title_zh": "整數唯一質因數分解",
      "reason_zh": "multiplicities。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數與映射",
      "reason_zh": "arithmetic function。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "mu(1)=1。",
      "square factor給0。",
      "squarefree sign為(-1)^k。"
    ],
    "common_misconception_zh": "只數distinct primes卻忽略square multiplicity，或把0分類。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "factorization不完整時不得輸出mu。",
    "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Möbius inversion。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-mobius-function.md 整理；2026-10-14 脈衝 R06（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"
  }
}