{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r04-wilson-theorem-0100",
  "pilot_number": "R0054-27725",
  "number": 27725,
  "family_id": "qf-mod49-wilson-boundary",
  "title": {
    "zh-Hant": "第 27725 題：Wilson定理",
    "en": "Question 27725: 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-0100",
    "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\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(fermat_confusion, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, 是, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_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, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, 是, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_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": 3,
      "skills": [
        "Wilson定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184100,
      "parameters": {
        "mode": "fermat_confusion",
        "n": 97,
        "domain": "Exact_Wilson_congruence",
        "operation": "wilson-boundary",
        "semantic_template": "wilson-boundary|{mode:\"fermat_confusion\",n:97}"
      }
    },
    "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-0100",
    "batch_id": "batch-20261015-advance-20261118-r04-wilson-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0a9934517116a1f6aed3acd34d94fd0275b63ae29f200df0da0735cb2da3125b",
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184100,
      "semantic_template": "wilson-boundary|{mode:\"fermat_confusion\",n:97}"
    },
    "fields": {
      "title": "Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate",
      "stem": "Use mode=fermat_confusion and exact integer n=97. 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, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, Yes, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_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, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, Yes, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_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\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use mode=fermat_confusion and exact integer n=97. 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, 97, 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, 97, 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, 97, 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, 97, 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, 97, 5, 24, 120]",
      "after": "derived = [1, 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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(23, 6)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 138",
      "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": [
          "23",
          "6"
        ],
        "output": "138",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product6"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 6, 23, 138]",
      "after": "derived = [1, 41, 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(41, 7)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 287",
      "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": [
          "41",
          "7"
        ],
        "output": "287",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product7"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 7, 41, 287]",
      "after": "derived = [2, 93, 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(93, 8)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 744",
      "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": [
          "93",
          "8"
        ],
        "output": "744",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product8"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 8, 93, 744]",
      "after": "derived = [7, 65, 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(65, 9)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 585",
      "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": [
          "65",
          "9"
        ],
        "output": "585",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product9"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 9, 65, 585]",
      "after": "derived = [6, 3, 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(3, 10)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 30",
      "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": [
          "3",
          "10"
        ],
        "output": "30",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product10"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 10, 3, 30]",
      "after": "derived = [0, 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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(30, 11)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 330",
      "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": [
          "30",
          "11"
        ],
        "output": "330",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product11"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 11, 30, 330]",
      "after": "derived = [3, 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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(39, 12)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 468",
      "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": [
          "39",
          "12"
        ],
        "output": "468",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product12"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 12, 39, 468]",
      "after": "derived = [4, 80, 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(80, 13)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 1040",
      "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": [
          "80",
          "13"
        ],
        "output": "1040",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product13"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 13, 80, 1040]",
      "after": "derived = [10, 70, 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(70, 14)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 980",
      "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": [
          "70",
          "14"
        ],
        "output": "980",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product14"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 14, 70, 980]",
      "after": "derived = [10, 10, 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(10, 15)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 150",
      "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": [
          "10",
          "15"
        ],
        "output": "150",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product15"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 15, 10, 150]",
      "after": "derived = [1, 53, 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(53, 16)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 848",
      "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": [
          "53",
          "16"
        ],
        "output": "848",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product16"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 16, 53, 848]",
      "after": "derived = [8, 72, 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(72, 17)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 1224",
      "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": [
          "72",
          "17"
        ],
        "output": "1224",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product17"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 17, 72, 1224]",
      "after": "derived = [12, 60, 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(60, 18)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 1080",
      "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": [
          "60",
          "18"
        ],
        "output": "1080",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product18"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 18, 60, 1080]",
      "after": "derived = [11, 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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(13, 19)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 247",
      "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": [
          "13",
          "19"
        ],
        "output": "247",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product19"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 19, 13, 247]",
      "after": "derived = [2, 53, 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(53, 20)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 1060",
      "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": [
          "53",
          "20"
        ],
        "output": "1060",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product20"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 20, 53, 1060]",
      "after": "derived = [10, 90, 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(90, 21)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 1890",
      "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": [
          "90",
          "21"
        ],
        "output": "1890",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product21"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 21, 90, 1890]",
      "after": "derived = [19, 47, 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(47, 22)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 1034",
      "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": [
          "47",
          "22"
        ],
        "output": "1034",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product22"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 22, 47, 1034]",
      "after": "derived = [10, 64, 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(64, 23)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 1472",
      "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": [
          "64",
          "23"
        ],
        "output": "1472",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product23"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 23, 64, 1472]",
      "after": "derived = [15, 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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(17, 24)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 408",
      "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": [
          "17",
          "24"
        ],
        "output": "408",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product24"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 24, 17, 408]",
      "after": "derived = [4, 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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(20, 25)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 500",
      "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": [
          "20",
          "25"
        ],
        "output": "500",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product25"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 25, 20, 500]",
      "after": "derived = [5, 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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(15, 26)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 390",
      "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": [
          "15",
          "26"
        ],
        "output": "390",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product26"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 26, 15, 390]",
      "after": "derived = [4, 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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(2, 27)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 54",
      "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": [
          "2",
          "27"
        ],
        "output": "54",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product27"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 27, 2, 54]",
      "after": "derived = [0, 54, 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(54, 28)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 1512",
      "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": [
          "54",
          "28"
        ],
        "output": "1512",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product28"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 28, 54, 1512]",
      "after": "derived = [15, 57, 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(57, 29)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 1653",
      "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": [
          "57",
          "29"
        ],
        "output": "1653",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product29"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 29, 57, 1653]",
      "after": "derived = [17, 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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(4, 30)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 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 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": [
          "4",
          "30"
        ],
        "output": "120",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product30"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 30, 4, 120]",
      "after": "derived = [1, 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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(23, 31)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 713",
      "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": [
          "23",
          "31"
        ],
        "output": "713",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product31"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 31, 23, 713]",
      "after": "derived = [7, 34, 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(34, 32)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 1088",
      "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": [
          "34",
          "32"
        ],
        "output": "1088",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product32"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 32, 34, 1088]",
      "after": "derived = [11, 21, 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(21, 33)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 693",
      "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": [
          "21",
          "33"
        ],
        "output": "693",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product33"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 33, 21, 693]",
      "after": "derived = [7, 14, 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(14, 34)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 476",
      "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": [
          "14",
          "34"
        ],
        "output": "476",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product34"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 34, 14, 476]",
      "after": "derived = [4, 88, 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(88, 35)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 3080",
      "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": [
          "88",
          "35"
        ],
        "output": "3080",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product35"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 35, 88, 3080]",
      "after": "derived = [31, 73, 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(73, 36)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 2628",
      "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": [
          "73",
          "36"
        ],
        "output": "2628",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product36"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 36, 73, 2628]",
      "after": "derived = [27, 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": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(9, 37)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 333",
      "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": [
          "9",
          "37"
        ],
        "output": "333",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product37"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 37, 9, 333]",
      "after": "derived = [3, 42, 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(42, 38)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 1596",
      "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": [
          "42",
          "38"
        ],
        "output": "1596",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product38"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 38, 42, 1596]",
      "after": "derived = [16, 44, 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(44, 39)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1716",
      "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": [
          "44",
          "39"
        ],
        "output": "1716",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product39"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 39, 44, 1716]",
      "after": "derived = [17, 67, 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(67, 40)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 2680",
      "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": [
          "67",
          "40"
        ],
        "output": "2680",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product40"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 40, 67, 2680]",
      "after": "derived = [27, 61, 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": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(61, 41)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 2501",
      "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 product41，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "61",
          "41"
        ],
        "output": "2501",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product41"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 41, 61, 2501]",
      "after": "derived = [25, 76, 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": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(76, 42)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 3192",
      "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 product42，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "76",
          "42"
        ],
        "output": "3192",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product42"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 42, 76, 3192]",
      "after": "derived = [32, 88, 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": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 := mul(88, 43)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 3784",
      "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 product43，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "88",
          "43"
        ],
        "output": "3784",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product43"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 43, 88, 3784]",
      "after": "derived = [39, 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": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(1, 44)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 44",
      "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 product44，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "44"
        ],
        "output": "44",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product44"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 44, 1, 44]",
      "after": "derived = [0, 44, 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": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(44, 45)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 1980",
      "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 product45，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "44",
          "45"
        ],
        "output": "1980",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product45"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 45, 44, 1980]",
      "after": "derived = [20, 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": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 := mul(40, 46)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 1840",
      "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 product46，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "46"
        ],
        "output": "1840",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product46"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 46, 40, 1840]",
      "after": "derived = [18, 94, 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": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(94, 47)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 4418",
      "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 product47，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "94",
          "47"
        ],
        "output": "4418",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product47"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 47, 94, 4418]",
      "after": "derived = [45, 53, 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": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(53, 48)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 2544",
      "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 product48，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "53",
          "48"
        ],
        "output": "2544",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product48"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 48, 53, 2544]",
      "after": "derived = [26, 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": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(22, 49)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 1078",
      "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 product49，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "49"
        ],
        "output": "1078",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product49"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 49, 22, 1078]",
      "after": "derived = [11, 11, 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": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 := mul(11, 50)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 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 product50，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "50"
        ],
        "output": "550",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product50"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 50, 11, 550]",
      "after": "derived = [5, 65, 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": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(65, 51)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 3315",
      "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 product51，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "65",
          "51"
        ],
        "output": "3315",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product51"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 51, 65, 3315]",
      "after": "derived = [34, 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": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(17, 52)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 884",
      "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 product52，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "52"
        ],
        "output": "884",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product52"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 52, 17, 884]",
      "after": "derived = [9, 11, 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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 := mul(11, 53)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 = 583",
      "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 product53，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "53"
        ],
        "output": "583",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product53"
      }
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 53, 11, 583]",
      "after": "derived = [6, 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": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 := mul(1, 54)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 = 54",
      "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 product54，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "54"
        ],
        "output": "54",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product54"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 54, 1, 54]",
      "after": "derived = [0, 54, 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": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 := mul(54, 55)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 = 2970",
      "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 product55，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "54",
          "55"
        ],
        "output": "2970",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product55"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 55, 54, 2970]",
      "after": "derived = [30, 60, 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": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 := mul(60, 56)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 = 3360",
      "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 product56，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "60",
          "56"
        ],
        "output": "3360",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product56"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 56, 60, 3360]",
      "after": "derived = [34, 62, 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": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 := mul(62, 57)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 = 3534",
      "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 product57，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "62",
          "57"
        ],
        "output": "3534",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product57"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 57, 62, 3534]",
      "after": "derived = [36, 42, 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": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 := mul(42, 58)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 = 2436",
      "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 product58，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "58"
        ],
        "output": "2436",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product58"
      }
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 58, 42, 2436]",
      "after": "derived = [25, 11, 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": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 := mul(11, 59)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 = 649",
      "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 product59，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "59"
        ],
        "output": "649",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product59"
      }
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 59, 11, 649]",
      "after": "derived = [6, 67, 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": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 := mul(67, 60)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 = 4020",
      "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 product60，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "67",
          "60"
        ],
        "output": "4020",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product60"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 60, 67, 4020]",
      "after": "derived = [41, 43, 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": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 := mul(43, 61)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 = 2623",
      "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 product61，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "43",
          "61"
        ],
        "output": "2623",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product61"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 61, 43, 2623]",
      "after": "derived = [27, 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": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 := mul(4, 62)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 = 248",
      "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 product62，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "62"
        ],
        "output": "248",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product62"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 62, 4, 248]",
      "after": "derived = [2, 54, 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": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 := mul(54, 63)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 = 3402",
      "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 product63，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "54",
          "63"
        ],
        "output": "3402",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product63"
      }
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 63, 54, 3402]",
      "after": "derived = [35, 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": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 := mul(7, 64)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 = 448",
      "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 product64，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "64"
        ],
        "output": "448",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product64"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 64, 7, 448]",
      "after": "derived = [4, 60, 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": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 := mul(60, 65)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 = 3900",
      "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 product65，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "60",
          "65"
        ],
        "output": "3900",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product65"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 65, 60, 3900]",
      "after": "derived = [40, 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": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 := mul(20, 66)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 = 1320",
      "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 product66，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "66"
        ],
        "output": "1320",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product66"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 66, 20, 1320]",
      "after": "derived = [13, 59, 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": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 := mul(59, 67)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 = 3953",
      "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 product67，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "59",
          "67"
        ],
        "output": "3953",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product67"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 67, 59, 3953]",
      "after": "derived = [40, 73, 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": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 := mul(73, 68)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 = 4964",
      "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 product68，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "73",
          "68"
        ],
        "output": "4964",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product68"
      }
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 68, 73, 4964]",
      "after": "derived = [51, 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": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 := mul(17, 69)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 = 1173",
      "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 product69，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "69"
        ],
        "output": "1173",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product69"
      }
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 69, 17, 1173]",
      "after": "derived = [12, 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": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 := mul(9, 70)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 = 630",
      "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 product70，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "70"
        ],
        "output": "630",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product70"
      }
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 70, 9, 630]",
      "after": "derived = [6, 48, 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": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 := mul(48, 71)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 = 3408",
      "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 product71，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "71"
        ],
        "output": "3408",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product71"
      }
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 71, 48, 3408]",
      "after": "derived = [35, 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": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 := mul(13, 72)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 = 936",
      "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 product72，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "72"
        ],
        "output": "936",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product72"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 72, 13, 936]",
      "after": "derived = [9, 63, 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": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 := mul(63, 73)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 = 4599",
      "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 product73，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "63",
          "73"
        ],
        "output": "4599",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product73"
      }
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 73, 63, 4599]",
      "after": "derived = [47, 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": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 := mul(40, 74)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 = 2960",
      "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 product74，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "74"
        ],
        "output": "2960",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product74"
      }
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 74, 40, 2960]",
      "after": "derived = [30, 50, 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": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 := mul(50, 75)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 = 3750",
      "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 product75，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "50",
          "75"
        ],
        "output": "3750",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product75"
      }
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 75, 50, 3750]",
      "after": "derived = [38, 64, 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": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 := mul(64, 76)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 = 4864",
      "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 product76，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "64",
          "76"
        ],
        "output": "4864",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product76"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 76, 64, 4864]",
      "after": "derived = [50, 14, 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": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 := mul(14, 77)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 = 1078",
      "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 product77，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "77"
        ],
        "output": "1078",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product77"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 77, 14, 1078]",
      "after": "derived = [11, 11, 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": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 := mul(11, 78)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 = 858",
      "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 product78，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "78"
        ],
        "output": "858",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product78"
      }
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 78, 11, 858]",
      "after": "derived = [8, 82, 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": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 := mul(82, 79)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 = 6478",
      "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 product79，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "82",
          "79"
        ],
        "output": "6478",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product79"
      }
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 79, 82, 6478]",
      "after": "derived = [66, 76, 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": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 := mul(76, 80)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 = 6080",
      "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 product80，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "76",
          "80"
        ],
        "output": "6080",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product80"
      }
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 80, 76, 6080]",
      "after": "derived = [62, 66, 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": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 := mul(66, 81)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 = 5346",
      "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 product81，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "66",
          "81"
        ],
        "output": "5346",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product81"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 81, 66, 5346]",
      "after": "derived = [55, 11, 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": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 := mul(11, 82)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 = 902",
      "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 product82，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "82"
        ],
        "output": "902",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product82"
      }
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 82, 11, 902]",
      "after": "derived = [9, 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": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 := mul(29, 83)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 = 2407",
      "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 product83，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "83"
        ],
        "output": "2407",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product83"
      }
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 83, 29, 2407]",
      "after": "derived = [24, 79, 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": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 := mul(79, 84)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 = 6636",
      "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 product84，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "79",
          "84"
        ],
        "output": "6636",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product84"
      }
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 84, 79, 6636]",
      "after": "derived = [68, 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": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 := mul(40, 85)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 = 3400",
      "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 product85，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "85"
        ],
        "output": "3400",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product85"
      }
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 85, 40, 3400]",
      "after": "derived = [35, 5, 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": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 := mul(5, 86)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 = 430",
      "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 product86，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "86"
        ],
        "output": "430",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product86"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 86, 5, 430]",
      "after": "derived = [4, 42, 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": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 := mul(42, 87)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 = 3654",
      "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 product87，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "87"
        ],
        "output": "3654",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product87"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 87, 42, 3654]",
      "after": "derived = [37, 65, 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": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 := mul(65, 88)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 = 5720",
      "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 product88，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "65",
          "88"
        ],
        "output": "5720",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product88"
      }
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 88, 65, 5720]",
      "after": "derived = [58, 94, 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": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 := mul(94, 89)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 = 8366",
      "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 product89，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "94",
          "89"
        ],
        "output": "8366",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product89"
      }
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 89, 94, 8366]",
      "after": "derived = [86, 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": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 := mul(24, 90)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 = 2160",
      "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 product90，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "90"
        ],
        "output": "2160",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product90"
      }
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 90, 24, 2160]",
      "after": "derived = [22, 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": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 := mul(26, 91)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 = 2366",
      "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 product91，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "91"
        ],
        "output": "2366",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product91"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 91, 26, 2366]",
      "after": "derived = [24, 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": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 := mul(38, 92)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 = 3496",
      "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 product92，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "92"
        ],
        "output": "3496",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product92"
      }
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 92, 38, 3496]",
      "after": "derived = [36, 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": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 := mul(4, 93)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 = 372",
      "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 product93，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "93"
        ],
        "output": "372",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product93"
      }
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 93, 4, 372]",
      "after": "derived = [3, 81, 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": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 := mul(81, 94)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 = 7614",
      "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 product94，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "81",
          "94"
        ],
        "output": "7614",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product94"
      }
    },
    {
      "id": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 94, 81, 7614]",
      "after": "derived = [78, 48, 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": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 := mul(48, 95)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 = 4560",
      "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 product95，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "95"
        ],
        "output": "4560",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product95"
      }
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 95, 48, 4560]",
      "after": "derived = [47, 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": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 := mul(1, 96)",
      "after": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 = 96",
      "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 product96，不以小數近似替代。",
          "en": "Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "96"
        ],
        "output": "96",
        "label": "Wilson boundary ORIGINAL prime ORIGINAL running factorial product96"
      }
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "source roles = [Wilson boundary ORIGINAL prime, 97, 96, 1, 96]",
      "after": "derived = [0, 96, 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": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "source roles = [97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96]",
      "after": "derived = [96, 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",
      97,
      [
        [
          1,
          1
        ],
        [
          2,
          2
        ],
        [
          3,
          6
        ],
        [
          4,
          24
        ],
        [
          5,
          23
        ],
        [
          6,
          41
        ],
        [
          7,
          93
        ],
        [
          8,
          65
        ],
        [
          9,
          3
        ],
        [
          10,
          30
        ],
        [
          11,
          39
        ],
        [
          12,
          80
        ],
        [
          13,
          70
        ],
        [
          14,
          10
        ],
        [
          15,
          53
        ],
        [
          16,
          72
        ],
        [
          17,
          60
        ],
        [
          18,
          13
        ],
        [
          19,
          53
        ],
        [
          20,
          90
        ],
        [
          21,
          47
        ],
        [
          22,
          64
        ],
        [
          23,
          17
        ],
        [
          24,
          20
        ],
        [
          25,
          15
        ],
        [
          26,
          2
        ],
        [
          27,
          54
        ],
        [
          28,
          57
        ],
        [
          29,
          4
        ],
        [
          30,
          23
        ],
        [
          31,
          34
        ],
        [
          32,
          21
        ],
        [
          33,
          14
        ],
        [
          34,
          88
        ],
        [
          35,
          73
        ],
        [
          36,
          9
        ],
        [
          37,
          42
        ],
        [
          38,
          44
        ],
        [
          39,
          67
        ],
        [
          40,
          61
        ],
        [
          41,
          76
        ],
        [
          42,
          88
        ],
        [
          43,
          1
        ],
        [
          44,
          44
        ],
        [
          45,
          40
        ],
        [
          46,
          94
        ],
        [
          47,
          53
        ],
        [
          48,
          22
        ],
        [
          49,
          11
        ],
        [
          50,
          65
        ],
        [
          51,
          17
        ],
        [
          52,
          11
        ],
        [
          53,
          1
        ],
        [
          54,
          54
        ],
        [
          55,
          60
        ],
        [
          56,
          62
        ],
        [
          57,
          42
        ],
        [
          58,
          11
        ],
        [
          59,
          67
        ],
        [
          60,
          43
        ],
        [
          61,
          4
        ],
        [
          62,
          54
        ],
        [
          63,
          7
        ],
        [
          64,
          60
        ],
        [
          65,
          20
        ],
        [
          66,
          59
        ],
        [
          67,
          73
        ],
        [
          68,
          17
        ],
        [
          69,
          9
        ],
        [
          70,
          48
        ],
        [
          71,
          13
        ],
        [
          72,
          63
        ],
        [
          73,
          40
        ],
        [
          74,
          50
        ],
        [
          75,
          64
        ],
        [
          76,
          14
        ],
        [
          77,
          11
        ],
        [
          78,
          82
        ],
        [
          79,
          76
        ],
        [
          80,
          66
        ],
        [
          81,
          11
        ],
        [
          82,
          29
        ],
        [
          83,
          79
        ],
        [
          84,
          40
        ],
        [
          85,
          5
        ],
        [
          86,
          42
        ],
        [
          87,
          65
        ],
        [
          88,
          94
        ],
        [
          89,
          24
        ],
        [
          90,
          26
        ],
        [
          91,
          38
        ],
        [
          92,
          4
        ],
        [
          93,
          81
        ],
        [
          94,
          48
        ],
        [
          95,
          1
        ],
        [
          96,
          96
        ]
      ],
      96,
      96,
      true,
      "Wilson:(n-1)!==-1_mod_97",
      "Fermat:a^(97-1)==1_mod_97_when_n_does_not_divide_a",
      false,
      "distinct_theorems_no_Fermat_power_claim_recomputed"
    ]
  },
  "conclusion": {
    "canonical_value": "(fermat_confusion, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, 是, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_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": 100,
      "original_line_sha256": "a75551fc2d8c192d2d3daee1dda6508f65a86ad5fd3a8b1c757e3d784943f4f0",
      "question_compact_sha256": "0a9934517116a1f6aed3acd34d94fd0275b63ae29f200df0da0735cb2da3125b",
      "english_row_compact_sha256": "d26478d8587bb978e79e3bff069eadd5abc978e452c929efdb281818e1cb79d6",
      "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": "6dfb3b566b1f87c8cb4fe311ca5bd52e25fe71ddd5a1f2f3dd033cf00746952a",
    "original_candidate_line_sha256": "6dfb3b566b1f87c8cb4fe311ca5bd52e25fe71ddd5a1f2f3dd033cf00746952a"
  },
  "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": 194,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9d74e5c0a21ac25b20b0036e800385f5ec8c140c1fc2a1f931f0edb249aa97f1",
      "arithmetic_steps": 96
    },
    "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."
  }
}