{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r05-fermat-little-theorem-0056",
  "pilot_number": "R0038-19856",
  "number": 19856,
  "family_id": "qf-mod37-flt-hypothesis-audit",
  "title": {
    "zh-Hant": "第 19856 題：Fermat小定理",
    "en": "Question 19856: An Inapplicable Theorem Need Not Mean a Failed Modular Test"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r05-fermat-little-theorem-0056",
    "batch_id": "batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fermat-little-theorem"
    ],
    "title_zh": "Fermat小定理",
    "stem_zh": "exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":9,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(9, -7, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)",
      "explanation_zh": "Wave5封閉層核對=(9, -7, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Fermat小定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod37-flt-hypothesis-audit",
      "seed": 202611065056,
      "parameters": {
        "modulus": 9,
        "a": -7,
        "domain": "Exact_modular_Fermat_little_theorem",
        "operation": "flt-hypothesis-audit",
        "semantic_template": "qf-mod37-flt-hypothesis-audit/{\"a\":-7,\"modulus\":9}"
      }
    },
    "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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、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-20261106-r05-fermat-little-theorem-0056",
    "batch_id": "batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "13090189f2f6f3bd14f9a1df50c14ecb3469561a86b281f80848c0aff9198b56",
      "family_id": "qf-mod37-flt-hypothesis-audit",
      "seed": 202611065056,
      "semantic_template": "qf-mod37-flt-hypothesis-audit/{\"a\":-7,\"modulus\":9}"
    },
    "fields": {
      "title": "An Inapplicable Theorem Need Not Mean a Failed Modular Test",
      "stem": "Use integer base a=-7 and modulus=9, normalized residues in [0,modulus). Exact integer square-and-multiply computes powers; trial division checks primality. Every equality of residues here is a congruence modulo the stated modulus, not an unrestricted integer equality. The coprime form requires prime modulus and an invertible base; the all-integer form for prime p also allows p to divide a. Never cancel a noninvertible residue. Give (modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(9, -7, No, Yes, No, 4, coprime_form_not_applicable_without_both_hypotheses)",
        "explanation": "Retain the original signed base, primality, coprimality, joint applicability and actual a^(modulus−1) residue separately. Failure of either hypothesis blocks the prime-coprime theorem, but does not force that residue to differ from1. The exact result, retaining the requested order, is (9, -7, No, Yes, No, 4, coprime_form_not_applicable_without_both_hypotheses)."
      },
      "skills": [
        "An Inapplicable Theorem Need Not Mean a Failed Modular Test"
      ],
      "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": "exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":9,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use integer base a=-7 and modulus=9, normalized residues in [0,modulus). Exact integer square-and-multiply computes powers; trial division checks primality. Every equality of residues here is a congruence modulo the stated modulus, not an unrestricted integer equality. The coprime form requires prime modulus and an invertible base; the all-integer form for prime p also allows p to divide a. Never cancel a noninvertible residue. Give (modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=flt-hypothesis-audit;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's 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": "ACTUAL integer modulus=9;complete trial range2..floor_sqrt=3",
      "after": "checked rows[divisor,quotient,remainder]=[[2, 4, 1], [3, 3, 0]];first exact factor=3;prime=否",
      "relation": "implies",
      "rule_id": "flt_primality_complete_trial_or_actual_factor_not_Fermat_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若找到2到sqrt範圍的真因數就是合數見證。若全部試除無因數，任意合數分解必至少一因數≤sqrt，故排除全部合成情形；2的試除範圍空仍合法。不是以一次Fermat測試證prime。",
        "en": "An exact factor in the trial range witnesses compositeness.If the entire range has no factor,any composite decomposition would require one at most sqrt(n),so all composite cases are excluded.The empty range for2 is valid.This is not a primality inference from a Fermat test."
      },
      "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": "original signed base=-7;positive modulus=9;use nonnegative gcd inputs=[7, 9]",
      "after": "ALL Euclidean rows[left,right,q,r]=[[7, 9, 0, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是",
      "relation": "implies",
      "rule_id": "flt_actual_gcd_coprime_separate_from_primality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "餘數替換保共同因數，最後非零值給gcd；base0也保gcd(0,m)=m。不把模數prime與base可逆當同一旗標。",
        "en": "Euclidean replacements preserve common divisors and give the terminal gcd,including gcd(0,m)=m.Primality of the modulus and invertibility of the base remain separate flags."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Fermat original signed base exact quotient := div(-7, 9)",
      "after": "Fermat original signed base exact quotient = -7/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original signed base exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original signed base exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-7",
          "9"
        ],
        "output": "-7/9",
        "label": "Fermat original signed base exact quotient"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Fermat original signed base floor quotient multiple := mul(-1, 9)",
      "after": "Fermat original signed base floor quotient multiple = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original signed base floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original signed base floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "9"
        ],
        "output": "-9",
        "label": "Fermat original signed base floor quotient multiple"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Fermat original signed base normalized remainder := sub(-7, -9)",
      "after": "Fermat original signed base normalized remainder = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original signed base normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original signed base normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-7",
          "-9"
        ],
        "output": "2",
        "label": "Fermat original signed base normalized remainder"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Fermat original signed base:signed integer value=-7;positive modulus=9;exact quotient=-7/9;floor q=-1",
      "after": "value=q*modulus+remainder;normalized remainder=2 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Fermat original modular exponent:original normalized base=2;original exponent=8;modulus=9",
      "after": "initialize accumulator1,current_base=base,remaining=exponent;invariant accumulator*current_base^remaining congruent to original power",
      "relation": "implies",
      "rule_id": "flt_modular_power_initial_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法單位元給0次方慣例；每步保持同一原始冪的同餘，不計龐大普通整數冪或混成實數等號。",
        "en": "The multiplicative identity gives the exponent0 convention.Each step preserves congruence to the same original power without constructing huge ordinary powers or replacing modular relations with real equalities."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Fermat original modular exponent bit0 current base square := mul(2, 2)",
      "after": "Fermat original modular exponent bit0 current base square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit0 current base square，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit0 current base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Fermat original modular exponent bit0 current base square"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Fermat original modular exponent bit0 squared base reduction exact quotient := div(4, 9)",
      "after": "Fermat original modular exponent bit0 squared base reduction exact quotient = 4/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit0 squared base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit0 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "9"
        ],
        "output": "4/9",
        "label": "Fermat original modular exponent bit0 squared base reduction exact quotient"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 9)",
      "after": "Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit0 squared base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit0 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "9"
        ],
        "output": "0",
        "label": "Fermat original modular exponent bit0 squared base reduction floor quotient multiple"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(4, 0)",
      "after": "Fermat original modular exponent bit0 squared base reduction normalized remainder = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit0 squared base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit0 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "Fermat original modular exponent bit0 squared base reduction normalized remainder"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Fermat original modular exponent bit0 squared base reduction:signed integer value=4;positive modulus=9;exact quotient=4/9;floor q=0",
      "after": "value=q*modulus+remainder;normalized remainder=4 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 2, 8];low bit=0",
      "after": "next[accumulator,base,exponent]=[1, 4, 4];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "flt_each_binary_power_step_exact_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偶次方換成平方底數，奇次方多取一個底數進accumulator；每次合法取模保同餘，剩餘非負exponent減半故終止。",
        "en": "Replace an even power by a squared base;an odd power also contributes one base factor to the accumulator.Exact reductions preserve congruence and halving the remaining nonnegative exponent guarantees termination."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Fermat original modular exponent bit1 current base square := mul(4, 4)",
      "after": "Fermat original modular exponent bit1 current base square = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit1 current base square，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit1 current base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "Fermat original modular exponent bit1 current base square"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Fermat original modular exponent bit1 squared base reduction exact quotient := div(16, 9)",
      "after": "Fermat original modular exponent bit1 squared base reduction exact quotient = 16/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit1 squared base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit1 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "16",
          "9"
        ],
        "output": "16/9",
        "label": "Fermat original modular exponent bit1 squared base reduction exact quotient"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(1, 9)",
      "after": "Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit1 squared base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit1 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "9"
        ],
        "output": "9",
        "label": "Fermat original modular exponent bit1 squared base reduction floor quotient multiple"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(16, 9)",
      "after": "Fermat original modular exponent bit1 squared base reduction normalized remainder = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit1 squared base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit1 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "16",
          "9"
        ],
        "output": "7",
        "label": "Fermat original modular exponent bit1 squared base reduction normalized remainder"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Fermat original modular exponent bit1 squared base reduction:signed integer value=16;positive modulus=9;exact quotient=16/9;floor q=1",
      "after": "value=q*modulus+remainder;normalized remainder=7 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 4, 4];low bit=0",
      "after": "next[accumulator,base,exponent]=[1, 7, 2];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "flt_each_binary_power_step_exact_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偶次方換成平方底數，奇次方多取一個底數進accumulator；每次合法取模保同餘，剩餘非負exponent減半故終止。",
        "en": "Replace an even power by a squared base;an odd power also contributes one base factor to the accumulator.Exact reductions preserve congruence and halving the remaining nonnegative exponent guarantees termination."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Fermat original modular exponent bit2 current base square := mul(7, 7)",
      "after": "Fermat original modular exponent bit2 current base square = 49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit2 current base square，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit2 current base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "7"
        ],
        "output": "49",
        "label": "Fermat original modular exponent bit2 current base square"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Fermat original modular exponent bit2 squared base reduction exact quotient := div(49, 9)",
      "after": "Fermat original modular exponent bit2 squared base reduction exact quotient = 49/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit2 squared base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit2 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "49",
          "9"
        ],
        "output": "49/9",
        "label": "Fermat original modular exponent bit2 squared base reduction exact quotient"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(5, 9)",
      "after": "Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit2 squared base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit2 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "9"
        ],
        "output": "45",
        "label": "Fermat original modular exponent bit2 squared base reduction floor quotient multiple"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(49, 45)",
      "after": "Fermat original modular exponent bit2 squared base reduction normalized remainder = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit2 squared base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit2 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "49",
          "45"
        ],
        "output": "4",
        "label": "Fermat original modular exponent bit2 squared base reduction normalized remainder"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Fermat original modular exponent bit2 squared base reduction:signed integer value=49;positive modulus=9;exact quotient=49/9;floor q=5",
      "after": "value=q*modulus+remainder;normalized remainder=4 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 7, 2];low bit=0",
      "after": "next[accumulator,base,exponent]=[1, 4, 1];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "flt_each_binary_power_step_exact_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偶次方換成平方底數，奇次方多取一個底數進accumulator；每次合法取模保同餘，剩餘非負exponent減半故終止。",
        "en": "Replace an even power by a squared base;an odd power also contributes one base factor to the accumulator.Exact reductions preserve congruence and halving the remaining nonnegative exponent guarantees termination."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Fermat original modular exponent bit3 accumulator product := mul(1, 4)",
      "after": "Fermat original modular exponent bit3 accumulator product = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 accumulator product，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 accumulator product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "4"
        ],
        "output": "4",
        "label": "Fermat original modular exponent bit3 accumulator product"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Fermat original modular exponent bit3 accumulator reduction exact quotient := div(4, 9)",
      "after": "Fermat original modular exponent bit3 accumulator reduction exact quotient = 4/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 accumulator reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "9"
        ],
        "output": "4/9",
        "label": "Fermat original modular exponent bit3 accumulator reduction exact quotient"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 9)",
      "after": "Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 accumulator reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "9"
        ],
        "output": "0",
        "label": "Fermat original modular exponent bit3 accumulator reduction floor quotient multiple"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(4, 0)",
      "after": "Fermat original modular exponent bit3 accumulator reduction normalized remainder = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 accumulator reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "Fermat original modular exponent bit3 accumulator reduction normalized remainder"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Fermat original modular exponent bit3 accumulator reduction:signed integer value=4;positive modulus=9;exact quotient=4/9;floor q=0",
      "after": "value=q*modulus+remainder;normalized remainder=4 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Fermat original modular exponent bit3 current base square := mul(4, 4)",
      "after": "Fermat original modular exponent bit3 current base square = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 current base square，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 current base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "Fermat original modular exponent bit3 current base square"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Fermat original modular exponent bit3 squared base reduction exact quotient := div(16, 9)",
      "after": "Fermat original modular exponent bit3 squared base reduction exact quotient = 16/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 squared base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "16",
          "9"
        ],
        "output": "16/9",
        "label": "Fermat original modular exponent bit3 squared base reduction exact quotient"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(1, 9)",
      "after": "Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 squared base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "9"
        ],
        "output": "9",
        "label": "Fermat original modular exponent bit3 squared base reduction floor quotient multiple"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(16, 9)",
      "after": "Fermat original modular exponent bit3 squared base reduction normalized remainder = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fermat original modular exponent bit3 squared base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Fermat original modular exponent bit3 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "16",
          "9"
        ],
        "output": "7",
        "label": "Fermat original modular exponent bit3 squared base reduction normalized remainder"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Fermat original modular exponent bit3 squared base reduction:signed integer value=16;positive modulus=9;exact quotient=16/9;floor q=1",
      "after": "value=q*modulus+remainder;normalized remainder=7 in[0,9)",
      "relation": "implies",
      "rule_id": "flt_exact_signed_modulo_not_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用正模數與向下整數商核實同餘類，負base也得到非負代表。餘數相等是mod此數的同餘，不是原整數都相等。",
        "en": "Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 4, 1];low bit=1",
      "after": "next[accumulator,base,exponent]=[4, 7, 0];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "flt_each_binary_power_step_exact_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偶次方換成平方底數，奇次方多取一個底數進accumulator；每次合法取模保同餘，剩餘非負exponent減半故終止。",
        "en": "Replace an even power by a squared base;an odd power also contributes one base factor to the accumulator.Exact reductions preserve congruence and halving the remaining nonnegative exponent guarantees termination."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Fermat original modular exponent:remaining exponent0;final normalized accumulator=4",
      "after": "EXACT original base^8 modulo9=4",
      "relation": "implies",
      "rule_id": "flt_modular_power_terminal_not_primality_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不變式在remaining0時只剩accumulator，得到原始冪的確切餘數；這個算術結果本身不證模數prime。",
        "en": "At remaining exponent0 the invariant leaves exactly the accumulator,giving the original power's residue.The arithmetic result alone does not certify a prime modulus."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ACTUAL modulus=9;ORIGINAL signed base=-7;prime=否;coprime=是;actual power residue=4",
      "after": "prime-coprime applicability=否;failed hypotheses do NOT force actual residue!=1;one successful residue does NOT prove prime",
      "relation": "implies",
      "rule_id": "flt_applicability_not_actual_power_failure_or_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "joint flag是兩個前提的合取，不是把算出的冪與1比較。合數也可能對某個base得到1；不適用不等於算術一定失敗，算術通過也不等於prime。",
        "en": "The joint flag conjoins the hypotheses rather than comparing the computed power with1.A composite may yield1 for a particular base.Inapplicability does not force arithmetic failure,and a passing power does not establish primality."
      },
      "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
    }
  ],
  "math_model": {
    "result": [
      9,
      -7,
      false,
      true,
      false,
      4,
      "coprime_form_not_applicable_without_both_hypotheses"
    ]
  },
  "conclusion": {
    "canonical_value": "(9, -7, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)",
    "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."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations/questions.jsonl",
      "batch_sha256": "88b1cb151738b4a35e4b932a85cbafdaf8ee4b9be0c49601150c9f12516f0e89",
      "original_line": 56,
      "original_line_sha256": "b87b6f42a71f94d16606626dc06cc38d28e1c2011e887b0fba781061722a4325",
      "question_compact_sha256": "13090189f2f6f3bd14f9a1df50c14ecb3469561a86b281f80848c0aff9198b56",
      "english_row_compact_sha256": "a2025584f277e6b188650ab8c50171d0b2b983bc1df7562bbef27709d9d8594e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-fermat-little-theorem": {
        "sha256": "8ba0aab812152c67f3df53e149427ab6d311dab16d274314d1ad46b4ee3114d7",
        "english_sha256": "26ee5d2d09455781b206bf52404c98ec56346bdfa0f20829131a82e3e7481aba",
        "original_path": "public/data/mko/source13-20261018-fermat-little-theorem.json",
        "english_path": "public/data/i18n/editions/mko-fermat-little-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11",
    "original_candidate_sha256": "a7cad13d3ce48c8f070facfaefd46bb60b45dd433646e51c63c21fbab49da397",
    "original_candidate_line_sha256": "a7cad13d3ce48c8f070facfaefd46bb60b45dd433646e51c63c21fbab49da397"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "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."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p39",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612",
      "question_first_witness_sha256": "c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 39,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e5a42f4b654a35e216ec1d2996aacfe55322083c7c1c0f302e39358b9d361900",
      "arithmetic_steps": 23
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}