{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-mobius-function-01",
  "mko_id": "mko-mobius-function",
  "title": {
    "zh-Hant": "完整因數分解與μ(30)",
    "en": "Complete factorization and mu(30)"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-mobius-function-01",
    "source": {
      "mko_id": "mko-mobius-function",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "8991dece49cf8a59e05519f80e840deb92afafba5fef912edc1a03ad30438f84",
      "english_sha256": "c602666c47bdf330fa90517c2baa6b3bb4811deb05ff4c991c801c6ba0793623"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "μ(30)=−1，μ(12)=0，μ(1)=1；30的全部divisor值加總為0。",
      "en": "mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "主例n=30，另核12與1；μ對平方因數為0、對k個distinct prime的squarefree整數為(−1)^k。",
          "en": "Main n=30, with 12 and 1; mu is zero for a repeated prime factor and (-1)^k for a squarefree integer with k distinct primes."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "30=2\\cdot15=2\\cdot3\\cdot5",
        "after_tex": "30=2^1 3^1 5^1",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整trial extraction先取2再3，餘5只需試2；2、3、5均prime，乘積回到30。",
          "en": "Complete trial extraction removes 2 then 3, leaving prime 5 after testing 2; the product returns 30."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "沒有未分類的剩餘因子，才能讀multiplicities。",
            "en": "No remaining factor is unclassified, so multiplicities can be read."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(e_2,e_3,e_5)=(1,1,1)",
        "after_tex": "30\\text{ squarefree},\\quad k=3",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部prime重數皆1，平方因數分支不觸發，distinct-prime count為3。",
          "en": "Every prime multiplicity is 1, so the square-factor branch does not apply and the distinct-prime count is 3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "先核重數再數種類，不能只數三種就略平方條件。",
            "en": "Check multiplicities before counting types; three distinct primes alone would not ensure squarefreeness."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "k=3",
        "after_tex": "\\mu(30)=(-1)^3=-1",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "按squarefree分支給符號−1。",
          "en": "The squarefree branch gives sign -1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "因數排序不改3個prime的個數或重數。",
            "en": "Reordering factors changes neither the count nor multiplicities."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "30=2^1 3^1 5^1",
        "after_tex": "D(30)=\\{1,2,3,5,6,10,15,30\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個指數各只能取0或1，2³=8種選擇全部列出positive divisors。",
          "en": "Each exponent is 0 or 1, and all 2^3=8 choices give the listed positive divisors."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "列表同時含1與30，並非只列prime或proper divisors。",
            "en": "The list includes both 1 and 30, not just prime or proper divisors."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S03",
          "S04"
        ],
        "before_tex": "d=1,2,3,5,6,10,15,30",
        "after_tex": "\\mu(d)=1,-1,-1,-1,1,1,1,-1\\text{ respectively}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個divisor皆由同一完整分解繼承重數；empty product對1給1。",
          "en": "Each divisor inherits complete multiplicities; the empty product for 1 gives 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "例如6有兩個prime給+1，30有三個給−1，不能用大小奇偶。",
            "en": "6 has two primes giving +1 and 30 three giving -1; numerical parity is irrelevant."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "\\sum_{d\\mid30}\\mu(d)",
        "after_tex": "1-1-1-1+1+1+1-1=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整八項加總給n>1的精確cross-check。",
          "en": "The complete eight-term sum gives the exact n>1 cross-check."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "這次有限核對只認證30，不取代一般divisor-sum定理。",
            "en": "This finite check certifies 30, not the general divisor-sum theorem."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "12=2^2\\cdot3",
        "after_tex": "2^2\\mid12\\Rightarrow\\mu(12)=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "2重複出現，故必先進平方因數分支，不能取(−1)²。",
          "en": "The repeated factor 2 forces the square-factor branch, not (-1)^2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "完整分解含重數2、1，distinct count2不足以分類。",
            "en": "Complete multiplicities are 2,1; distinct count 2 alone is insufficient."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S05",
          "S07"
        ],
        "before_tex": "D(1)=\\{1\\},\\quad\\mu(1)=1",
        "after_tex": "\\sum_{d\\mid1}\\mu(d)=1;\\quad n\\le0\\text{ outside domain}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "n=1另保留empty prime product與sum1；0、負整數不擴張定義。",
          "en": "Retain the empty prime product and divisor sum 1 at n=1; zero and negatives are outside this definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "本稿不定義 μ(0)，因此不能依本定義賦值為 0；0 不在此稿的分類定義域內。",
            "en": "mu(0) is not assigned zero; it is outside the classification domain."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "μ(30)=−1，μ(12)=0，μ(1)=1；30的全部divisor值加總為0。",
      "en": "mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "μ(12)=1因有兩種prime。",
          "en": "mu(12)=1 because there are two prime types."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "2²整除12，故值0。",
          "en": "2^2 divides 12, forcing zero."
        }
      },
      {
        "claim": {
          "zh-Hant": "μ(1)=−1。",
          "en": "mu(1)=-1."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "空prime product的k=0，(−1)^0=1。",
          "en": "The empty prime product has k=0, so (-1)^0=1."
        }
      },
      {
        "claim": {
          "zh-Hant": "μ(0)=0依本定義。",
          "en": "mu(0)=0 under this definition."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "domain是positive integers，0不分類。",
          "en": "The domain is positive integers; zero is unclassified."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設精確整數算術、帶餘除法、整除與gcd的定義；同餘等式始終綁定明示模數。",
          "en": "Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "A complete unique-prime-factorization certificate supplies multiplicities and the distinct-prime count, independently of factor order. The exact divisor-sum cross-check is sum over d dividing n of mu(d)=1 for n=1 and 0 otherwise. Zero and negative integers are outside the domain. Incomplete trial division does not justify classification."
        },
        "source": {
          "mko_id": "mko-mobius-function",
          "sha256": "8991dece49cf8a59e05519f80e840deb92afafba5fef912edc1a03ad30438f84",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "product",
        "lhs": "2*3*5",
        "rhs": "30",
        "step": "S01"
      },
      {
        "id": "sum",
        "lhs": "1-1-1-1+1+1+1-1",
        "rhs": "0",
        "step": "S06"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "正整數完整UPF與完整divisor cross-check，未完成分解不分類。",
      "en": "Complete positive-integer factorization and divisor cross-checks; incomplete factors give no classification."
    },
    "md_symbols": [
      [
        "\\mu",
        "Mobius函數",
        "Mobius function"
      ],
      [
        "D(n)",
        "全部positive divisors",
        "all positive divisors"
      ]
    ],
    "md_error_controls": [
      {
        "title": "重數先行 / Multiplicity first",
        "zh": "任何prime重數≥2就給0，distinct count不再決定符號。",
        "en": "Any prime multiplicity at least 2 forces zero; distinct count then does not determine a sign."
      },
      {
        "title": "完整分解 / Complete factorization",
        "zh": "未試到sqrt(remaining)前不可把剩餘數直接當prime。",
        "en": "Do not label a remainder prime before trial division reaches its square-root bound."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "3505226a6366d7f0fe4ff0cdea84d0829765aebda7ef89862b6364d7922604cd",
    "original_source_sha256": "59850a6096a974b7810ab0b6adb6274aa72c4c80f42e39bd8ed038c2ccad9727",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S156",
      "BR142-P053"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}