{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0043",
  "pilot_number": "P1000-0943",
  "number": 943,
  "family_id": "qf-mod47-morder-power-first-return",
  "title": {
    "zh-Hant": "第 943 題：模n乘法階",
    "en": "Question 943: The First Positive Power Returning to One Defines the Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0043",
    "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-multiplicative-order-mod-n"
    ],
    "title_zh": "模n乘法階",
    "stem_zh": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":20,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)",
      "explanation_zh": "Wave6第十層核對=([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "模n乘法階"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-morder-power-first-return",
      "seed": 202611167043,
      "parameters": {
        "n": 20,
        "a": 3,
        "domain": "Exact_multiplicative_order_mod_n",
        "operation": "morder-power-first-return",
        "semantic_template": "qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":20}"
      }
    },
    "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、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0043",
    "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ff04d6c9c15688008198c14de4dd6ee876765773c439dddfd21f41a1a9fe9df7",
      "family_id": "qf-mod47-morder-power-first-return",
      "seed": 202611167043,
      "semantic_template": "qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":20}"
    },
    "fields": {
      "title": "The First Positive Power Returning to One Defines the Order",
      "stem": "Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)",
        "explanation": "Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."
      },
      "skills": [
        "The First Positive Power Returning to One Defines the Order"
      ],
      "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": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":20,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=20; a=3",
      "after": "order is defined here only for n≥2 and units modulo n",
      "relation": "implies",
      "rule_id": "order_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定本節點的定義域；加法階數與乘法階數不能混用。",
        "en": "Fix this node's domain first; do not confuse additive and multiplicative order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "a=3",
      "after": "normalized residue=3; gcd(3,20)=1",
      "relation": "implies",
      "rule_id": "normalize_and_qualify",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把負代表元也正規化到[0,n)，再檢查互質。",
        "en": "Normalize even a negative representative into[0,n), then test coprimality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "gcd(residue,n)=1; units=[1, 3, 7, 9, 11, 13, 17, 19]",
      "after": "phi(20)=8",
      "relation": "implies",
      "rule_id": "complete_unit_enumeration",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一枚舉0到n−1的互質類，phi是完整unit載體的大小。",
        "en": "Enumerate every coprime class from0 through n−1; phi is the complete unit-carrier size."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "power product at k=1 := mul(1, 3)",
      "after": "power product at k=1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前一個冪的代表乘以同一正規化底數。",
        "en": "Multiply the preceding power representative by the same normalized base."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 power product at k=1，不以小數近似替代。",
          "en": "Recompute power product at k=1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "power product at k=1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "product=3",
      "after": "a^1 mod 20=3",
      "relation": "implies",
      "rule_id": "reduce_next_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此步餘數才是完整的下一冪代表。",
        "en": "The remainder is the next full power representative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "power product at k=2 := mul(3, 3)",
      "after": "power product at k=2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前一個冪的代表乘以同一正規化底數。",
        "en": "Multiply the preceding power representative by the same normalized base."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 power product at k=2，不以小數近似替代。",
          "en": "Recompute power product at k=2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "power product at k=2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "product=9",
      "after": "a^2 mod 20=9",
      "relation": "implies",
      "rule_id": "reduce_next_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此步餘數才是完整的下一冪代表。",
        "en": "The remainder is the next full power representative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "power product at k=3 := mul(9, 3)",
      "after": "power product at k=3 = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前一個冪的代表乘以同一正規化底數。",
        "en": "Multiply the preceding power representative by the same normalized base."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 power product at k=3，不以小數近似替代。",
          "en": "Recompute power product at k=3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "3"
        ],
        "output": "27",
        "label": "power product at k=3"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "product=27",
      "after": "a^3 mod 20=7",
      "relation": "implies",
      "rule_id": "reduce_next_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此步餘數才是完整的下一冪代表。",
        "en": "The remainder is the next full power representative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "power product at k=4 := mul(7, 3)",
      "after": "power product at k=4 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前一個冪的代表乘以同一正規化底數。",
        "en": "Multiply the preceding power representative by the same normalized base."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 power product at k=4，不以小數近似替代。",
          "en": "Recompute power product at k=4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "3"
        ],
        "output": "21",
        "label": "power product at k=4"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "product=21",
      "after": "a^4 mod 20=1",
      "relation": "implies",
      "rule_id": "reduce_next_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此步餘數才是完整的下一冪代表。",
        "en": "The remainder is the next full power representative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "all positive k<=4: [[1, 3], [2, 9], [3, 7], [4, 1]]",
      "after": "first positive return to1 occurs at k=4",
      "relation": "implies",
      "rule_id": "least_positive_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已完整檢查所有更小正指數均未回到1；k=0的單位元不計入階數。",
        "en": "Every smaller positive exponent has been checked not to return to1; the identity at exponent0 is not the order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。",
          "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          1,
          3
        ],
        [
          2,
          9
        ],
        [
          3,
          7
        ],
        [
          4,
          1
        ]
      ],
      4,
      true,
      true,
      "order_is_the_first_positive_return"
    ]
  },
  "conclusion": {
    "canonical_value": "([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "n≥2 且 a 正規化後與 n 互質；階數是第一個回到 1 的正指數。",
    "en": "Require n≥2 and a normalized residue coprime to n. The order is the first positive exponent returning to 1."
  },
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    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}