{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0003",
  "pilot_number": "R0043-22103",
  "number": 22103,
  "family_id": "qf-mod41-mobinv-forward-divisor-transform",
  "title": {
    "zh-Hant": "第 22103 題：有限因數格上的Möbius反演",
    "en": "Question 22103: The Forward Transform Sums over All Positive Divisors"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0003",
    "batch_id": "batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mobius-inversion-finite-divisors"
    ],
    "title_zh": "有限因數格上的Möbius反演",
    "stem_zh": "positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":3,\"a\":3,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)",
      "explanation_zh": "Wave6第四層核對=(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限因數格上的Möbius反演"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-mobinv-forward-divisor-transform",
      "seed": 202611106003,
      "parameters": {
        "n": 3,
        "a": 3,
        "b": -10,
        "domain": "Exact_finite_divisor_Mobius_inversion",
        "operation": "mobinv-forward-divisor-transform",
        "semantic_template": "qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-10,\"n\":3}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0003",
    "batch_id": "batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-mod41-mobinv-forward-divisor-transform",
      "seed": 202611106003,
      "semantic_template": "qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-10,\"n\":3}"
    },
    "fields": {
      "title": "The Forward Transform Sums over All Positive Divisors",
      "stem": "Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)",
        "explanation": "List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)."
      },
      "skills": [
        "The Forward Transform Sums over All Positive Divisors"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":3,\"a\":3,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=mobinv-forward-divisor-transform;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[n, divisor_value_rows, F_n, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Mobius inversion ORIGINAL forward:actual positive input=3;check complete integer range1..3",
      "after": "ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor := mul(3, 1)",
      "after": "Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b := sum(3, -10)",
      "after": "Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b = -7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "-10"
        ],
        "output": "-7",
        "label": "Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor := mul(3, 3)",
      "after": "Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b := sum(9, -10)",
      "after": "Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "-10"
        ],
        "output": "-1",
        "label": "Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Mobius inversion ORIGINAL forward full forward divisor sum := sum(-7, -1)",
      "after": "Mobius inversion ORIGINAL forward full forward divisor sum = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL forward full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL forward full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7",
          "-1"
        ],
        "output": "-8",
        "label": "Mobius inversion ORIGINAL forward full forward divisor sum"
      }
    }
  ],
  "math_model": {
    "result": [
      3,
      [
        [
          1,
          -7
        ],
        [
          3,
          -1
        ]
      ],
      -8,
      "sum_over_all_positive_divisors"
    ]
  },
  "conclusion": {
    "canonical_value": "(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
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