{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r04-wilson-theorem-0088",
  "pilot_number": "R0054-27713",
  "number": 27713,
  "family_id": "qf-mod49-wilson-boundary",
  "title": {
    "zh-Hant": "第 27713 題：Wilson定理",
    "en": "Question 27713: Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r04-wilson-theorem-0088",
    "batch_id": "batch-20261015-advance-20261118-r04-wilson-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-wilson-theorem"
    ],
    "title_zh": "Wilson定理",
    "stem_zh": "n>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-boundary；資料={\"mode\":\"fermat_confusion\",\"n\":41}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, 是, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, 否, distinct_theorems_no_Fermat_power_claim_recomputed)",
      "explanation_zh": "Wave6第十二層核對=(fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, 是, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, 否, distinct_theorems_no_Fermat_power_claim_recomputed)；依完整明示資料獨立重算；p=2只有一個self-inverse residue 1=-1；composite instances與finite testing不取代converse proof，Wilson不等於Fermat。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Wilson定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184088,
      "parameters": {
        "mode": "fermat_confusion",
        "n": 41,
        "domain": "Exact_Wilson_congruence",
        "operation": "wilson-boundary",
        "semantic_template": "wilson-boundary|{mode:\"fermat_confusion\",n:41}"
      }
    },
    "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；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、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-20261118-r04-wilson-theorem-0088",
    "batch_id": "batch-20261015-advance-20261118-r04-wilson-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2941fea468674d19dcf312694b2a8218f03237c14acbbeed85d14ac6a8318165",
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184088,
      "semantic_template": "wilson-boundary|{mode:\"fermat_confusion\",n:41}"
    },
    "fields": {
      "title": "Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate",
      "stem": "Use mode=fermat_confusion and exact integer n=41. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, Yes, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)",
        "explanation": "Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, Yes, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)."
      },
      "skills": [
        "Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate"
      ],
      "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>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-boundary；資料={\"mode\":\"fermat_confusion\",\"n\":41}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use mode=fermat_confusion and exact integer n=41. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [wilson-boundary]",
      "after": "derived = [[mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。",
        "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 := mul(1, 1)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product1，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 1, 1, 1]",
      "after": "derived = [0, 1, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 := mul(1, 2)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product2，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 2, 1, 2]",
      "after": "derived = [0, 2, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 := mul(2, 3)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product3，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product3"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 3, 2, 6]",
      "after": "derived = [0, 6, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 := mul(6, 4)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product4，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "4"
        ],
        "output": "24",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product4"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 4, 6, 24]",
      "after": "derived = [0, 24, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 := mul(24, 5)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product5，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "5"
        ],
        "output": "120",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product5"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 5, 24, 120]",
      "after": "derived = [2, 38, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(38, 6)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 228",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product6，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "6"
        ],
        "output": "228",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 6, 38, 228]",
      "after": "derived = [5, 23, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(23, 7)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 161",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product7，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "7"
        ],
        "output": "161",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product7"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 7, 23, 161]",
      "after": "derived = [3, 38, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(38, 8)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 304",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product8，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "8"
        ],
        "output": "304",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product8"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 8, 38, 304]",
      "after": "derived = [7, 17, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(17, 9)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 153",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product9，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "9"
        ],
        "output": "153",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product9"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 9, 17, 153]",
      "after": "derived = [3, 30, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(30, 10)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 300",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product10，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "10"
        ],
        "output": "300",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product10"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 10, 30, 300]",
      "after": "derived = [7, 13, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(13, 11)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 143",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product11，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "11"
        ],
        "output": "143",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 11, 13, 143]",
      "after": "derived = [3, 20, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(20, 12)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 240",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product12，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "12"
        ],
        "output": "240",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 12, 20, 240]",
      "after": "derived = [5, 35, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(35, 13)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 455",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product13，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "35",
          "13"
        ],
        "output": "455",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product13"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 13, 35, 455]",
      "after": "derived = [11, 4, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(4, 14)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 56",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product14，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "14"
        ],
        "output": "56",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product14"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 14, 4, 56]",
      "after": "derived = [1, 15, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(15, 15)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product15，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "15"
        ],
        "output": "225",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product15"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 15, 15, 225]",
      "after": "derived = [5, 20, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(20, 16)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 320",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product16，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "16"
        ],
        "output": "320",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product16"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 16, 20, 320]",
      "after": "derived = [7, 33, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(33, 17)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 561",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product17，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "17"
        ],
        "output": "561",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product17"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 17, 33, 561]",
      "after": "derived = [13, 28, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(28, 18)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 504",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product18，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28",
          "18"
        ],
        "output": "504",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product18"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 18, 28, 504]",
      "after": "derived = [12, 12, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(12, 19)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 228",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product19，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "19"
        ],
        "output": "228",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 19, 12, 228]",
      "after": "derived = [5, 23, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(23, 20)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 460",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product20，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "20"
        ],
        "output": "460",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product20"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 20, 23, 460]",
      "after": "derived = [11, 9, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(9, 21)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 189",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product21，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "21"
        ],
        "output": "189",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product21"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 21, 9, 189]",
      "after": "derived = [4, 25, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(25, 22)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 550",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product22，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "22"
        ],
        "output": "550",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product22"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 22, 25, 550]",
      "after": "derived = [13, 17, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(17, 23)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 391",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product23，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "23"
        ],
        "output": "391",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product23"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 23, 17, 391]",
      "after": "derived = [9, 22, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(22, 24)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 528",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product24，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "24"
        ],
        "output": "528",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 24, 22, 528]",
      "after": "derived = [12, 36, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(36, 25)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 900",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product25，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "25"
        ],
        "output": "900",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 25, 36, 900]",
      "after": "derived = [21, 39, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(39, 26)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 1014",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product26，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "26"
        ],
        "output": "1014",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 26, 39, 1014]",
      "after": "derived = [24, 30, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(30, 27)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 810",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product27，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "27"
        ],
        "output": "810",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 27, 30, 810]",
      "after": "derived = [19, 31, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(31, 28)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 868",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product28，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "31",
          "28"
        ],
        "output": "868",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product28"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 28, 31, 868]",
      "after": "derived = [21, 7, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(7, 29)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 203",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product29，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "29"
        ],
        "output": "203",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product29"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 29, 7, 203]",
      "after": "derived = [4, 39, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(39, 30)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 1170",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product30，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "30"
        ],
        "output": "1170",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 30, 39, 1170]",
      "after": "derived = [28, 22, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(22, 31)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 682",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product31，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "31"
        ],
        "output": "682",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 31, 22, 682]",
      "after": "derived = [16, 26, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(26, 32)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 832",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product32，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "32"
        ],
        "output": "832",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product32"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 32, 26, 832]",
      "after": "derived = [20, 12, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(12, 33)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 396",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product33，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "33"
        ],
        "output": "396",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product33"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 33, 12, 396]",
      "after": "derived = [9, 27, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(27, 34)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 918",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product34，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "34"
        ],
        "output": "918",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product34"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 34, 27, 918]",
      "after": "derived = [22, 16, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 := mul(16, 35)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 560",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product35，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "35"
        ],
        "output": "560",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product35"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 35, 16, 560]",
      "after": "derived = [13, 27, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 := mul(27, 36)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 972",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product36，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "36"
        ],
        "output": "972",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product36"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 36, 27, 972]",
      "after": "derived = [23, 29, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(29, 37)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1073",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product37，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "37"
        ],
        "output": "1073",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 37, 29, 1073]",
      "after": "derived = [26, 7, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(7, 38)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 266",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product38，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "38"
        ],
        "output": "266",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product38"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 38, 7, 266]",
      "after": "derived = [6, 20, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 := mul(20, 39)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 780",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product39，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "39"
        ],
        "output": "780",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product39"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 39, 20, 780]",
      "after": "derived = [19, 1, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 := mul(1, 40)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Wilson boundary ORIGINAL prime ORIGINAL running factorial product40，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "40"
        ],
        "output": "40",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product40"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 41, 40, 1, 40]",
      "after": "derived = [0, 40, 是, 是]",
      "relation": "implies",
      "rule_id": "Wilson_EACH_running_factorial_exact_Euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐k保前residue與完整乘積，再以唯一Euclidean餘數縮回0..n-1；累積不變量是value≡k! mod n，不把table最後一項當未算的假設。",
        "en": "Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "source roles = [41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40]",
      "after": "derived = [40, 是, 否]",
      "relation": "implies",
      "rule_id": "Wilson_and_Fermat_DISTINCT_named_statements_NO_unprovided_a_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "Wilson談(n-1)!；Fermat談n不整除a時a^(n-1)。此分支未給a也未重算其冪，不能把named comparison當同一定理或額外數值測試。",
        "en": "Wilson concerns(n-1)!;Fermat concerns a^(n-1) when n does not divide a.This branch supplies no a or power computation.The named comparison neither identifies the theorems nor adds an executed power test."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "fermat_confusion",
      41,
      [
        [
          1,
          1
        ],
        [
          2,
          2
        ],
        [
          3,
          6
        ],
        [
          4,
          24
        ],
        [
          5,
          38
        ],
        [
          6,
          23
        ],
        [
          7,
          38
        ],
        [
          8,
          17
        ],
        [
          9,
          30
        ],
        [
          10,
          13
        ],
        [
          11,
          20
        ],
        [
          12,
          35
        ],
        [
          13,
          4
        ],
        [
          14,
          15
        ],
        [
          15,
          20
        ],
        [
          16,
          33
        ],
        [
          17,
          28
        ],
        [
          18,
          12
        ],
        [
          19,
          23
        ],
        [
          20,
          9
        ],
        [
          21,
          25
        ],
        [
          22,
          17
        ],
        [
          23,
          22
        ],
        [
          24,
          36
        ],
        [
          25,
          39
        ],
        [
          26,
          30
        ],
        [
          27,
          31
        ],
        [
          28,
          7
        ],
        [
          29,
          39
        ],
        [
          30,
          22
        ],
        [
          31,
          26
        ],
        [
          32,
          12
        ],
        [
          33,
          27
        ],
        [
          34,
          16
        ],
        [
          35,
          27
        ],
        [
          36,
          29
        ],
        [
          37,
          7
        ],
        [
          38,
          20
        ],
        [
          39,
          1
        ],
        [
          40,
          40
        ]
      ],
      40,
      40,
      true,
      "Wilson:(n-1)!==-1_mod_41",
      "Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a",
      false,
      "distinct_theorems_no_Fermat_power_claim_recomputed"
    ]
  },
  "conclusion": {
    "canonical_value": "(fermat_confusion, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, 是, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_when_n_does_not_divide_a, 否, distinct_theorems_no_Fermat_power_claim_recomputed)",
    "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-20261015-advance-20261118-r04-wilson-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r04-wilson-theorem-foundations/questions.jsonl",
      "batch_sha256": "d1aac7811ce398b2e86d71092a3480737f06aa76137b9fe7b65b1fe52487ff92",
      "original_line": 88,
      "original_line_sha256": "3726c1252f49fdeffd30efc2dbb07de88b7a1a0163cd60b2f6af414fcad7fd0f",
      "question_compact_sha256": "2941fea468674d19dcf312694b2a8218f03237c14acbbeed85d14ac6a8318165",
      "english_row_compact_sha256": "737e56fa18b0bd4bd8efa2801c450cff01239aef862e7e7c66af0a67227a5ae0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r04-wilson-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r04-wilson-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-wilson-theorem": {
        "sha256": "32b4a0da9e832eb6c7c0c33d723d8f66608b7013dcd990ad19fa2596126d9eb8",
        "english_sha256": "0ea5bedaf95ddb7a9d900b474d7830c9ce429c691d48ecd945d3920fcd1d2b48",
        "original_path": "public/data/mko/wilson-theorem.json",
        "english_path": "public/data/i18n/editions/mko-wilson-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8",
    "original_candidate_sha256": "3c27f46a7efceabcbc9175fc515dd4a18e5170dd2685f6c601341ef2546bf33b",
    "original_candidate_line_sha256": "3c27f46a7efceabcbc9175fc515dd4a18e5170dd2685f6c601341ef2546bf33b"
  },
  "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-20261026-010-p55",
    "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": "0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54",
      "question_first_witness_sha256": "09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 82,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "dde2afe3f91ffd4efb74a32f87e989235b0e3a7ac21769bc697dfbfabdd18137",
      "arithmetic_steps": 40
    },
    "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."
  }
}