{
  "$schema": "../../../schemas/mko-v0.5.schema.json",
  "schema_version": "mko-v0.5",
  "id": "mko-unique-prime-factorization",
  "semantic_family_id": "sf-unique-prime-factorization",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "整數唯一質因數分解"
  },
  "summary": {
    "zh-Hant": "每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。"
  },
  "statement": {
    "zh-Hant": "existence以well-founded/induction與完整trial division小切片驗證；uniqueness保留Euclid lemma witness p|ab implies p|a or p|b，而非只比較少數答案。整數版分離negative sign，±1為units與empty product，0沒有此prime factorization。"
  },
  "formula": {
    "tex": "n=\\prod_{i=1}^{r}p_i^{e_i}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>n</mi><mo>=</mo><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>r</mi></munderover><msup><msub><mi>p</mi><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub></msup></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "n",
        "glyph": "n"
      },
      "rhs": {
        "type": "finite_product",
        "index": {
          "type": "symbol",
          "name": "i",
          "glyph": "i"
        },
        "lower": {
          "type": "number",
          "value": 1
        },
        "upper": {
          "type": "symbol",
          "name": "r",
          "glyph": "r"
        },
        "body": {
          "type": "power",
          "base": {
            "type": "subscript",
            "base": {
              "type": "symbol",
              "name": "p",
              "glyph": "p"
            },
            "subscript": {
              "type": "symbol",
              "name": "i",
              "glyph": "i"
            }
          },
          "exponent": {
            "type": "subscript",
            "base": {
              "type": "symbol",
              "name": "e",
              "glyph": "e"
            },
            "subscript": {
              "type": "symbol",
              "name": "i",
              "glyph": "i"
            }
          }
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-source13",
      "version": "0.5.0",
      "source_sha256": "ffd4e84f587e5bd84881a04d2f376634b325f50503a2b4ca527c9ac9a34a8e29"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "主定理域為正整數n>1。"
    },
    {
      "id": "a2",
      "zh-Hant": "bounded computational companions完整搜尋到sqrt(remaining)並exact驗prime與product。"
    }
  ],
  "symbols": [
    {
      "token": "p_i",
      "role": "prime_factor",
      "role_zh": "第 i 個質因數",
      "scope": "representative_formula"
    },
    {
      "token": "e_i",
      "role": "prime_exponent",
      "role_zh": "第 i 個正整數指數",
      "scope": "representative_formula"
    },
    {
      "token": "p",
      "role": "ordered_prime_family",
      "role_zh": "有序互異質因數族",
      "scope": "representative_formula"
    },
    {
      "token": "e",
      "role": "positive_exponent_family",
      "role_zh": "正整數指數族",
      "scope": "representative_formula"
    },
    {
      "token": "i",
      "role": "bound_factor_index",
      "role_zh": "有限乘積的束縛索引",
      "scope": "product_bound_index"
    },
    {
      "token": "r",
      "role": "number_of_distinct_prime_factors",
      "role_zh": "互異質因數個數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "positive_integer_to_factor",
      "role_zh": "待分解正整數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-prime-number",
      "title_zh": "自然數質數",
      "reason_zh": "prime factors。"
    },
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除",
      "reason_zh": "divisibility。"
    },
    {
      "id": "mko-mathematical-induction",
      "title_zh": "數學歸納法",
      "reason_zh": "existence。"
    },
    {
      "id": "mko-euclidean-algorithm",
      "title_zh": "Euclidean演算法",
      "reason_zh": "Euclid lemma支撐。"
    },
    {
      "id": "mko-gcd-coprime",
      "title_zh": "最大公因數與互質",
      "reason_zh": "prime divisibility。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "existence與uniqueness分開。",
      "uniqueness比較prime multisets忽略order。",
      "units、negative sign與zero需另分支。",
      "記號對齊候選：主定理 n>1 時，r>=1，p_1<...<p_r 為互異質數且每個 e_i 為正整數；乘積涵蓋所有質因數，不限質數冪。原有 ±1 的單位／空乘積及 0 排除界線仍保留。此代表式不自行證明存在唯一性。"
    ],
    "common_misconception_zh": "只找到一個分解便稱已證unique，或把0與units套入n>1版本。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "trial division僅在完整bounded range時構成certificate。",
    "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p-adic valuation與Möbius function。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-unique-prime-factorization.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-08 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"
  }
}