{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-mobius-inversion-finite-divisors",
  "semantic_family_id": "sf-mobius-inversion-finite-divisors",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限因數格上的Möbius反演"
  },
  "summary": {
    "zh-Hant": "若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。"
  },
  "statement": {
    "zh-Hant": "positive integer n上，若F(n)=sum_{d|n}f(d)，則f(n)=sum_{d|n}mu(d)F(n/d)。等價index形式可用mu(n/d)F(d)，但不得混合。必須枚舉全部positive divisors並由完整UPF取得mu；n=1與mu(1)=1保留。這是divisor-poset convolution，不是d≤n。"
  },
  "formula": {
    "tex": "f(n)=I(F,n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>I</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "I",
          "glyph": "I"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "F",
            "glyph": "F"
          },
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2ae5392cfab0d006e7c01a81d41a12f9040c37adc70c25bd7dcab6d0856ae160"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "n為positive safe integer且divisor enumeration完整。"
    },
    {
      "id": "a2",
      "zh-Hant": "題族使用exact integer arithmetic function f(k)=ak+b。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "original_arithmetic_function",
      "role_zh": "原算術函數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "positive_integer",
      "role_zh": "正整數",
      "scope": "representative_formula"
    },
    {
      "token": "I",
      "role": "mobius_inversion_operator",
      "role_zh": "Möbius反演算子",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "divisor_sum_transform",
      "role_zh": "因數和轉換",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-mobius-function",
      "title_zh": "Möbius函數",
      "reason_zh": "mu weights。"
    },
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除關係",
      "reason_zh": "complete divisor set。"
    },
    {
      "id": "mko-counting-principle",
      "title_zh": "基本計數原理",
      "reason_zh": "finite regrouping。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "forward transform用all divisors。",
      "inverse保持d與n/d indexing一致。",
      "roundtrip包含n=1。"
    ],
    "common_misconception_zh": "只加prime/proper divisors，或把d|n誤成d≤n。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "0、negative與incomplete factorization/divisor list不在本輪正向domain。",
    "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Legendre symbol與Euler criterion。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-mobius-inversion-finite-divisors.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-10 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"
  }
}