{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r06-finite-field-0021",
  "pilot_number": "R0066-33646",
  "number": 33646,
  "family_id": "qf-mod31-fielddeep-characteristic",
  "title": {
    "zh-Hant": "第 33646 題：有限域深度",
    "en": "Question 33646: Characteristic Is the Additive Order of One"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r06-finite-field-0021",
    "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-field"
    ],
    "title_zh": "有限域深度",
    "stem_zh": "完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":73}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)",
      "explanation_zh": "Wave4收束核對=(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限域深度"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-fielddeep-characteristic",
      "seed": 202610316021,
      "parameters": {
        "p": 73,
        "domain": "Exact_prime_field_Zp",
        "operation": "fielddeep-characteristic",
        "semantic_template": "qf-mod31-fielddeep-characteristic/{\"p\":73}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261031-r06-finite-field-0021",
    "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "be31defa1b64c0717c09b1b927219bf5ef532d8cdebbdf81fbcd4a322fcd8892",
      "family_id": "qf-mod31-fielddeep-characteristic",
      "seed": 202610316021,
      "semantic_template": "qf-mod31-fielddeep-characteristic/{\"p\":73}"
    },
    "fields": {
      "title": "Characteristic Is the Additive Order of One",
      "stem": "Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.",
      "answer": {
        "kind": "expression",
        "value": "(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)",
        "explanation": "The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)."
      },
      "skills": [
        "Characteristic Is the Additive Order of One"
      ],
      "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": "完整prime field Z_p運算。操作=fielddeep-characteristic；資料={\"p\":73}。請依 [p,successive_sums_of_one,characteristic,certificate] 欄序回答。",
        "en": "Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod31-fielddeep-characteristic, [domain, operation, p]]",
      "after": "derived = [fielddeep-characteristic, Exact_prime_field_Zp]",
      "relation": "implies",
      "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。",
        "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [73, [2, 3, 4, 5, 6, 7, 8]]",
      "after": "derived = [[1, 1, 1, 3, 1, 3, 1], 是]",
      "relation": "implies",
      "rule_id": "Field_PRIME_complete_trial_divisor_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整試除2到floor(sqrt(p))無因數；若p合成必有其中因數。2也合法。只限定本題Z_p，不把一般有限域都說成prime order或把有限環都說成域。",
        "en": "Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [successive sum of one k=0, 0, 73]",
      "after": "derived = [0, 0]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [successive sum of one k=1, 1, 73]",
      "after": "derived = [0, 1]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [successive sum of one k=2, 2, 73]",
      "after": "derived = [0, 2]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [successive sum of one k=3, 3, 73]",
      "after": "derived = [0, 3]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [successive sum of one k=4, 4, 73]",
      "after": "derived = [0, 4]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [successive sum of one k=5, 5, 73]",
      "after": "derived = [0, 5]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [successive sum of one k=6, 6, 73]",
      "after": "derived = [0, 6]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [successive sum of one k=7, 7, 73]",
      "after": "derived = [0, 7]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [successive sum of one k=8, 8, 73]",
      "after": "derived = [0, 8]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [successive sum of one k=9, 9, 73]",
      "after": "derived = [0, 9]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [successive sum of one k=10, 10, 73]",
      "after": "derived = [0, 10]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [successive sum of one k=11, 11, 73]",
      "after": "derived = [0, 11]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [successive sum of one k=12, 12, 73]",
      "after": "derived = [0, 12]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [successive sum of one k=13, 13, 73]",
      "after": "derived = [0, 13]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [successive sum of one k=14, 14, 73]",
      "after": "derived = [0, 14]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [successive sum of one k=15, 15, 73]",
      "after": "derived = [0, 15]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [successive sum of one k=16, 16, 73]",
      "after": "derived = [0, 16]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [successive sum of one k=17, 17, 73]",
      "after": "derived = [0, 17]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [successive sum of one k=18, 18, 73]",
      "after": "derived = [0, 18]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [successive sum of one k=19, 19, 73]",
      "after": "derived = [0, 19]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [successive sum of one k=20, 20, 73]",
      "after": "derived = [0, 20]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [successive sum of one k=21, 21, 73]",
      "after": "derived = [0, 21]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "source roles = [successive sum of one k=22, 22, 73]",
      "after": "derived = [0, 22]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [successive sum of one k=23, 23, 73]",
      "after": "derived = [0, 23]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [successive sum of one k=24, 24, 73]",
      "after": "derived = [0, 24]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [successive sum of one k=25, 25, 73]",
      "after": "derived = [0, 25]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "source roles = [successive sum of one k=26, 26, 73]",
      "after": "derived = [0, 26]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [successive sum of one k=27, 27, 73]",
      "after": "derived = [0, 27]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "source roles = [successive sum of one k=28, 28, 73]",
      "after": "derived = [0, 28]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [successive sum of one k=29, 29, 73]",
      "after": "derived = [0, 29]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "source roles = [successive sum of one k=30, 30, 73]",
      "after": "derived = [0, 30]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [successive sum of one k=31, 31, 73]",
      "after": "derived = [0, 31]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [successive sum of one k=32, 32, 73]",
      "after": "derived = [0, 32]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "source roles = [successive sum of one k=33, 33, 73]",
      "after": "derived = [0, 33]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "source roles = [successive sum of one k=34, 34, 73]",
      "after": "derived = [0, 34]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [successive sum of one k=35, 35, 73]",
      "after": "derived = [0, 35]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "source roles = [successive sum of one k=36, 36, 73]",
      "after": "derived = [0, 36]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [successive sum of one k=37, 37, 73]",
      "after": "derived = [0, 37]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "source roles = [successive sum of one k=38, 38, 73]",
      "after": "derived = [0, 38]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "source roles = [successive sum of one k=39, 39, 73]",
      "after": "derived = [0, 39]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "source roles = [successive sum of one k=40, 40, 73]",
      "after": "derived = [0, 40]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "source roles = [successive sum of one k=41, 41, 73]",
      "after": "derived = [0, 41]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "source roles = [successive sum of one k=42, 42, 73]",
      "after": "derived = [0, 42]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "source roles = [successive sum of one k=43, 43, 73]",
      "after": "derived = [0, 43]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "source roles = [successive sum of one k=44, 44, 73]",
      "after": "derived = [0, 44]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "source roles = [successive sum of one k=45, 45, 73]",
      "after": "derived = [0, 45]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "source roles = [successive sum of one k=46, 46, 73]",
      "after": "derived = [0, 46]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "source roles = [successive sum of one k=47, 47, 73]",
      "after": "derived = [0, 47]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "source roles = [successive sum of one k=48, 48, 73]",
      "after": "derived = [0, 48]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [successive sum of one k=49, 49, 73]",
      "after": "derived = [0, 49]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "source roles = [successive sum of one k=50, 50, 73]",
      "after": "derived = [0, 50]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "source roles = [successive sum of one k=51, 51, 73]",
      "after": "derived = [0, 51]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "source roles = [successive sum of one k=52, 52, 73]",
      "after": "derived = [0, 52]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "source roles = [successive sum of one k=53, 53, 73]",
      "after": "derived = [0, 53]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "source roles = [successive sum of one k=54, 54, 73]",
      "after": "derived = [0, 54]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "source roles = [successive sum of one k=55, 55, 73]",
      "after": "derived = [0, 55]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [successive sum of one k=56, 56, 73]",
      "after": "derived = [0, 56]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "source roles = [successive sum of one k=57, 57, 73]",
      "after": "derived = [0, 57]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [successive sum of one k=58, 58, 73]",
      "after": "derived = [0, 58]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "source roles = [successive sum of one k=59, 59, 73]",
      "after": "derived = [0, 59]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "source roles = [successive sum of one k=60, 60, 73]",
      "after": "derived = [0, 60]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "source roles = [successive sum of one k=61, 61, 73]",
      "after": "derived = [0, 61]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "source roles = [successive sum of one k=62, 62, 73]",
      "after": "derived = [0, 62]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "source roles = [successive sum of one k=63, 63, 73]",
      "after": "derived = [0, 63]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "source roles = [successive sum of one k=64, 64, 73]",
      "after": "derived = [0, 64]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "source roles = [successive sum of one k=65, 65, 73]",
      "after": "derived = [0, 65]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "source roles = [successive sum of one k=66, 66, 73]",
      "after": "derived = [0, 66]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "source roles = [successive sum of one k=67, 67, 73]",
      "after": "derived = [0, 67]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "source roles = [successive sum of one k=68, 68, 73]",
      "after": "derived = [0, 68]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "source roles = [successive sum of one k=69, 69, 73]",
      "after": "derived = [0, 69]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [successive sum of one k=70, 70, 73]",
      "after": "derived = [0, 70]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "source roles = [successive sum of one k=71, 71, 73]",
      "after": "derived = [0, 71]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "source roles = [successive sum of one k=72, 72, 73]",
      "after": "derived = [0, 72]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "source roles = [successive sum of one k=73, 73, 73]",
      "after": "derived = [1, 0]",
      "relation": "implies",
      "rule_id": "Finite_MOD_euclidean_quotient_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder<n，負代表元也正規化；不是浮點餘數。",
        "en": "Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "source roles = [73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0]]",
      "after": "derived = [73]",
      "relation": "implies",
      "rule_id": "Field_CHARACTERISTIC_first_POSITIVE_additive_return",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從0到p個1逐項相加取residue；1..p−1都非0，p才回0，所以characteristic=p。k=0不算正return，這不是乘法元素階。",
        "en": "List sums of0 through p copies of1.Residues1..p−1 are nonzero and p returns0,so the FIRST POSITIVE additive return is p.Zero is not a positive return;this is not multiplicative order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      73,
      [
        0,
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21,
        22,
        23,
        24,
        25,
        26,
        27,
        28,
        29,
        30,
        31,
        32,
        33,
        34,
        35,
        36,
        37,
        38,
        39,
        40,
        41,
        42,
        43,
        44,
        45,
        46,
        47,
        48,
        49,
        50,
        51,
        52,
        53,
        54,
        55,
        56,
        57,
        58,
        59,
        60,
        61,
        62,
        63,
        64,
        65,
        66,
        67,
        68,
        69,
        70,
        71,
        72,
        0
      ],
      73,
      "additive_order_of_one"
    ]
  },
  "conclusion": {
    "canonical_value": "(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r06-finite-field-foundations/questions.jsonl",
      "batch_sha256": "c74dfa9dad93779660bbfe9658232a134f33227465402a30690edbb933af30b5",
      "original_line": 21,
      "original_line_sha256": "637270f743094562ad048a10dbff0794b510c0d31ea8f4b96286fbb4bc3edbc1",
      "question_compact_sha256": "be31defa1b64c0717c09b1b927219bf5ef532d8cdebbdf81fbcd4a322fcd8892",
      "english_row_compact_sha256": "c8b49d78a8f506d17b9f442b90bfb9e6da84060686e582c6927744e7a91fa9a8",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r06-finite-field-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r06-finite-field-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-field": {
        "sha256": "a0f82af13b19a69bc7a3bab914cede8b8bae9db78543343f63b0b4086a684c5e",
        "english_sha256": "6afed76061d7bb9aae31a76426475e249103042ef12a966308ab75b556c16593",
        "original_path": "public/data/mko/finite-field.json",
        "english_path": "public/data/i18n/editions/mko-finite-field.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310",
    "original_candidate_sha256": "8008a51347ba1044afb78f8cf2a01c18f7e156a97843fe699a0f46b1805768e3",
    "original_candidate_line_sha256": "8008a51347ba1044afb78f8cf2a01c18f7e156a97843fe699a0f46b1805768e3"
  },
  "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-20261026-012-p67",
    "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": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267",
      "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 77,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6701787e2814eaf6df2629d9bcf351a2b2596bdebae20676ce4e1facd3c37513",
      "arithmetic_steps": 0
    },
    "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."
  }
}