{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261111-r06-quadratic-residue-0025",
  "pilot_number": "R0045-23400",
  "number": 23400,
  "family_id": "qf-mod42-quadratic-square-table",
  "title": {
    "zh-Hant": "第 23400 題：二次剩餘",
    "en": "Question 23400: The Complete Nonzero Square Table Has Half as Many Distinct Residues"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261111-r06-quadratic-residue-0025",
    "batch_id": "batch-20261014-advance-20261111-r06-quadratic-residue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quadratic-residue"
    ],
    "title_zh": "二次剩餘",
    "stem_zh": "odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, 是)",
      "explanation_zh": "Wave6第五層核對=(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "二次剩餘"
      ]
    },
    "variant": {
      "family_id": "qf-mod42-quadratic-square-table",
      "seed": 202611116025,
      "parameters": {
        "p": 101,
        "domain": "Odd_prime_nonzero_quadratic_residue",
        "operation": "quadratic-square-table",
        "semantic_template": "qf-mod42-quadratic-square-table/{\"p\":101}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261111-r06-quadratic-residue-0025",
    "batch_id": "batch-20261014-advance-20261111-r06-quadratic-residue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "831c7dc9730e28dfb5a264c5c0072a14fd27584d10cb1ceadb8b2fcc6a32a8a2",
      "family_id": "qf-mod42-quadratic-square-table",
      "seed": 202611116025,
      "semantic_template": "qf-mod42-quadratic-square-table/{\"p\":101}"
    },
    "fields": {
      "title": "The Complete Nonzero Square Table Has Half as Many Distinct Residues",
      "stem": "Use bounded odd prime p=101 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal).",
      "answer": {
        "kind": "expression",
        "value": "(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, Yes)",
        "explanation": "List [root,square class] in increasing nonzero-root order, deduplicate square classes increasingly, and compare their count with (p−1)/2. The zero class is not added. The exact result, retaining the requested order, is (101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, Yes)."
      },
      "skills": [
        "The Complete Nonzero Square Table Has Half as Many Distinct Residues"
      ],
      "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": "odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use bounded odd prime p=101 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=quadratic-square-table;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。",
        "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Quadratic ORIGINAL modulus:ORIGINAL integer=101;complete possible trial divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10]",
      "after": "ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 1], [5, 1], [6, 5], [7, 3], [8, 5], [9, 2], [10, 1]];prime=是;a composite>1 has a factor at most its square root",
      "relation": "implies",
      "rule_id": "finite_prime_ALL_trial_divisors_through_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整平方根範圍核整除；prime2的範圍為空而通過，0/1不通過。不能由少數取樣、平方表或Euler判準代替此資格。",
        "en": "Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Quadratic ORIGINAL nonzero root1 square := mul(1, 1)",
      "after": "Quadratic ORIGINAL nonzero root1 square = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root1 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Quadratic ORIGINAL nonzero root1 square"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ORIGINAL nonzero root x=1 in1..100;actual square=1",
      "after": "Euclidean square=0*101+1;nonzero square class=1;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Quadratic ORIGINAL nonzero root2 square := mul(2, 2)",
      "after": "Quadratic ORIGINAL nonzero root2 square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root2 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root2 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Quadratic ORIGINAL nonzero root2 square"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ORIGINAL nonzero root x=2 in1..100;actual square=4",
      "after": "Euclidean square=0*101+4;nonzero square class=4;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Quadratic ORIGINAL nonzero root3 square := mul(3, 3)",
      "after": "Quadratic ORIGINAL nonzero root3 square = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root3 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root3 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Quadratic ORIGINAL nonzero root3 square"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL nonzero root x=3 in1..100;actual square=9",
      "after": "Euclidean square=0*101+9;nonzero square class=9;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Quadratic ORIGINAL nonzero root4 square := mul(4, 4)",
      "after": "Quadratic ORIGINAL nonzero root4 square = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root4 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root4 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "Quadratic ORIGINAL nonzero root4 square"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ORIGINAL nonzero root x=4 in1..100;actual square=16",
      "after": "Euclidean square=0*101+16;nonzero square class=16;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Quadratic ORIGINAL nonzero root5 square := mul(5, 5)",
      "after": "Quadratic ORIGINAL nonzero root5 square = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root5 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root5 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "Quadratic ORIGINAL nonzero root5 square"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ORIGINAL nonzero root x=5 in1..100;actual square=25",
      "after": "Euclidean square=0*101+25;nonzero square class=25;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Quadratic ORIGINAL nonzero root6 square := mul(6, 6)",
      "after": "Quadratic ORIGINAL nonzero root6 square = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root6 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root6 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "6"
        ],
        "output": "36",
        "label": "Quadratic ORIGINAL nonzero root6 square"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL nonzero root x=6 in1..100;actual square=36",
      "after": "Euclidean square=0*101+36;nonzero square class=36;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Quadratic ORIGINAL nonzero root7 square := mul(7, 7)",
      "after": "Quadratic ORIGINAL nonzero root7 square = 49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root7 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root7 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "7"
        ],
        "output": "49",
        "label": "Quadratic ORIGINAL nonzero root7 square"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ORIGINAL nonzero root x=7 in1..100;actual square=49",
      "after": "Euclidean square=0*101+49;nonzero square class=49;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Quadratic ORIGINAL nonzero root8 square := mul(8, 8)",
      "after": "Quadratic ORIGINAL nonzero root8 square = 64",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root8 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root8 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "8"
        ],
        "output": "64",
        "label": "Quadratic ORIGINAL nonzero root8 square"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL nonzero root x=8 in1..100;actual square=64",
      "after": "Euclidean square=0*101+64;nonzero square class=64;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Quadratic ORIGINAL nonzero root9 square := mul(9, 9)",
      "after": "Quadratic ORIGINAL nonzero root9 square = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root9 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root9 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "9"
        ],
        "output": "81",
        "label": "Quadratic ORIGINAL nonzero root9 square"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ORIGINAL nonzero root x=9 in1..100;actual square=81",
      "after": "Euclidean square=0*101+81;nonzero square class=81;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Quadratic ORIGINAL nonzero root10 square := mul(10, 10)",
      "after": "Quadratic ORIGINAL nonzero root10 square = 100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root10 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root10 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "10"
        ],
        "output": "100",
        "label": "Quadratic ORIGINAL nonzero root10 square"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "ORIGINAL nonzero root x=10 in1..100;actual square=100",
      "after": "Euclidean square=0*101+100;nonzero square class=100;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Quadratic ORIGINAL nonzero root11 square := mul(11, 11)",
      "after": "Quadratic ORIGINAL nonzero root11 square = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root11 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root11 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "11"
        ],
        "output": "121",
        "label": "Quadratic ORIGINAL nonzero root11 square"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ORIGINAL nonzero root x=11 in1..100;actual square=121",
      "after": "Euclidean square=1*101+20;nonzero square class=20;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Quadratic ORIGINAL nonzero root12 square := mul(12, 12)",
      "after": "Quadratic ORIGINAL nonzero root12 square = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root12 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root12 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "12"
        ],
        "output": "144",
        "label": "Quadratic ORIGINAL nonzero root12 square"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ORIGINAL nonzero root x=12 in1..100;actual square=144",
      "after": "Euclidean square=1*101+43;nonzero square class=43;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Quadratic ORIGINAL nonzero root13 square := mul(13, 13)",
      "after": "Quadratic ORIGINAL nonzero root13 square = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root13 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root13 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "13"
        ],
        "output": "169",
        "label": "Quadratic ORIGINAL nonzero root13 square"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ORIGINAL nonzero root x=13 in1..100;actual square=169",
      "after": "Euclidean square=1*101+68;nonzero square class=68;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Quadratic ORIGINAL nonzero root14 square := mul(14, 14)",
      "after": "Quadratic ORIGINAL nonzero root14 square = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root14 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root14 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "14"
        ],
        "output": "196",
        "label": "Quadratic ORIGINAL nonzero root14 square"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "ORIGINAL nonzero root x=14 in1..100;actual square=196",
      "after": "Euclidean square=1*101+95;nonzero square class=95;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Quadratic ORIGINAL nonzero root15 square := mul(15, 15)",
      "after": "Quadratic ORIGINAL nonzero root15 square = 225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root15 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root15 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "15"
        ],
        "output": "225",
        "label": "Quadratic ORIGINAL nonzero root15 square"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL nonzero root x=15 in1..100;actual square=225",
      "after": "Euclidean square=2*101+23;nonzero square class=23;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Quadratic ORIGINAL nonzero root16 square := mul(16, 16)",
      "after": "Quadratic ORIGINAL nonzero root16 square = 256",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root16 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root16 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "16"
        ],
        "output": "256",
        "label": "Quadratic ORIGINAL nonzero root16 square"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ORIGINAL nonzero root x=16 in1..100;actual square=256",
      "after": "Euclidean square=2*101+54;nonzero square class=54;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Quadratic ORIGINAL nonzero root17 square := mul(17, 17)",
      "after": "Quadratic ORIGINAL nonzero root17 square = 289",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root17 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root17 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "17"
        ],
        "output": "289",
        "label": "Quadratic ORIGINAL nonzero root17 square"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "ORIGINAL nonzero root x=17 in1..100;actual square=289",
      "after": "Euclidean square=2*101+87;nonzero square class=87;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Quadratic ORIGINAL nonzero root18 square := mul(18, 18)",
      "after": "Quadratic ORIGINAL nonzero root18 square = 324",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root18 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root18 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "18"
        ],
        "output": "324",
        "label": "Quadratic ORIGINAL nonzero root18 square"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ORIGINAL nonzero root x=18 in1..100;actual square=324",
      "after": "Euclidean square=3*101+21;nonzero square class=21;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Quadratic ORIGINAL nonzero root19 square := mul(19, 19)",
      "after": "Quadratic ORIGINAL nonzero root19 square = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root19 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root19 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "19"
        ],
        "output": "361",
        "label": "Quadratic ORIGINAL nonzero root19 square"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ORIGINAL nonzero root x=19 in1..100;actual square=361",
      "after": "Euclidean square=3*101+58;nonzero square class=58;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Quadratic ORIGINAL nonzero root20 square := mul(20, 20)",
      "after": "Quadratic ORIGINAL nonzero root20 square = 400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root20 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root20 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "20"
        ],
        "output": "400",
        "label": "Quadratic ORIGINAL nonzero root20 square"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ORIGINAL nonzero root x=20 in1..100;actual square=400",
      "after": "Euclidean square=3*101+97;nonzero square class=97;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Quadratic ORIGINAL nonzero root21 square := mul(21, 21)",
      "after": "Quadratic ORIGINAL nonzero root21 square = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root21 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root21 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "21"
        ],
        "output": "441",
        "label": "Quadratic ORIGINAL nonzero root21 square"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL nonzero root x=21 in1..100;actual square=441",
      "after": "Euclidean square=4*101+37;nonzero square class=37;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Quadratic ORIGINAL nonzero root22 square := mul(22, 22)",
      "after": "Quadratic ORIGINAL nonzero root22 square = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root22 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root22 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "22"
        ],
        "output": "484",
        "label": "Quadratic ORIGINAL nonzero root22 square"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "ORIGINAL nonzero root x=22 in1..100;actual square=484",
      "after": "Euclidean square=4*101+80;nonzero square class=80;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Quadratic ORIGINAL nonzero root23 square := mul(23, 23)",
      "after": "Quadratic ORIGINAL nonzero root23 square = 529",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root23 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root23 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "23"
        ],
        "output": "529",
        "label": "Quadratic ORIGINAL nonzero root23 square"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "ORIGINAL nonzero root x=23 in1..100;actual square=529",
      "after": "Euclidean square=5*101+24;nonzero square class=24;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Quadratic ORIGINAL nonzero root24 square := mul(24, 24)",
      "after": "Quadratic ORIGINAL nonzero root24 square = 576",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root24 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root24 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "24"
        ],
        "output": "576",
        "label": "Quadratic ORIGINAL nonzero root24 square"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "ORIGINAL nonzero root x=24 in1..100;actual square=576",
      "after": "Euclidean square=5*101+71;nonzero square class=71;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Quadratic ORIGINAL nonzero root25 square := mul(25, 25)",
      "after": "Quadratic ORIGINAL nonzero root25 square = 625",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root25 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root25 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "25"
        ],
        "output": "625",
        "label": "Quadratic ORIGINAL nonzero root25 square"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "ORIGINAL nonzero root x=25 in1..100;actual square=625",
      "after": "Euclidean square=6*101+19;nonzero square class=19;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Quadratic ORIGINAL nonzero root26 square := mul(26, 26)",
      "after": "Quadratic ORIGINAL nonzero root26 square = 676",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root26 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root26 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "26"
        ],
        "output": "676",
        "label": "Quadratic ORIGINAL nonzero root26 square"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "ORIGINAL nonzero root x=26 in1..100;actual square=676",
      "after": "Euclidean square=6*101+70;nonzero square class=70;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Quadratic ORIGINAL nonzero root27 square := mul(27, 27)",
      "after": "Quadratic ORIGINAL nonzero root27 square = 729",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root27 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root27 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "27"
        ],
        "output": "729",
        "label": "Quadratic ORIGINAL nonzero root27 square"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "ORIGINAL nonzero root x=27 in1..100;actual square=729",
      "after": "Euclidean square=7*101+22;nonzero square class=22;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Quadratic ORIGINAL nonzero root28 square := mul(28, 28)",
      "after": "Quadratic ORIGINAL nonzero root28 square = 784",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root28 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root28 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28",
          "28"
        ],
        "output": "784",
        "label": "Quadratic ORIGINAL nonzero root28 square"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "ORIGINAL nonzero root x=28 in1..100;actual square=784",
      "after": "Euclidean square=7*101+77;nonzero square class=77;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Quadratic ORIGINAL nonzero root29 square := mul(29, 29)",
      "after": "Quadratic ORIGINAL nonzero root29 square = 841",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root29 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root29 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "29"
        ],
        "output": "841",
        "label": "Quadratic ORIGINAL nonzero root29 square"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "ORIGINAL nonzero root x=29 in1..100;actual square=841",
      "after": "Euclidean square=8*101+33;nonzero square class=33;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Quadratic ORIGINAL nonzero root30 square := mul(30, 30)",
      "after": "Quadratic ORIGINAL nonzero root30 square = 900",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root30 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root30 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "30"
        ],
        "output": "900",
        "label": "Quadratic ORIGINAL nonzero root30 square"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "ORIGINAL nonzero root x=30 in1..100;actual square=900",
      "after": "Euclidean square=8*101+92;nonzero square class=92;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Quadratic ORIGINAL nonzero root31 square := mul(31, 31)",
      "after": "Quadratic ORIGINAL nonzero root31 square = 961",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root31 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root31 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "31",
          "31"
        ],
        "output": "961",
        "label": "Quadratic ORIGINAL nonzero root31 square"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "ORIGINAL nonzero root x=31 in1..100;actual square=961",
      "after": "Euclidean square=9*101+52;nonzero square class=52;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Quadratic ORIGINAL nonzero root32 square := mul(32, 32)",
      "after": "Quadratic ORIGINAL nonzero root32 square = 1024",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root32 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root32 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "32"
        ],
        "output": "1024",
        "label": "Quadratic ORIGINAL nonzero root32 square"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "ORIGINAL nonzero root x=32 in1..100;actual square=1024",
      "after": "Euclidean square=10*101+14;nonzero square class=14;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Quadratic ORIGINAL nonzero root33 square := mul(33, 33)",
      "after": "Quadratic ORIGINAL nonzero root33 square = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root33 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root33 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "33"
        ],
        "output": "1089",
        "label": "Quadratic ORIGINAL nonzero root33 square"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "ORIGINAL nonzero root x=33 in1..100;actual square=1089",
      "after": "Euclidean square=10*101+79;nonzero square class=79;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Quadratic ORIGINAL nonzero root34 square := mul(34, 34)",
      "after": "Quadratic ORIGINAL nonzero root34 square = 1156",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root34 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root34 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "34",
          "34"
        ],
        "output": "1156",
        "label": "Quadratic ORIGINAL nonzero root34 square"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "ORIGINAL nonzero root x=34 in1..100;actual square=1156",
      "after": "Euclidean square=11*101+45;nonzero square class=45;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "Quadratic ORIGINAL nonzero root35 square := mul(35, 35)",
      "after": "Quadratic ORIGINAL nonzero root35 square = 1225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root35 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root35 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "35",
          "35"
        ],
        "output": "1225",
        "label": "Quadratic ORIGINAL nonzero root35 square"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "ORIGINAL nonzero root x=35 in1..100;actual square=1225",
      "after": "Euclidean square=12*101+13;nonzero square class=13;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "Quadratic ORIGINAL nonzero root36 square := mul(36, 36)",
      "after": "Quadratic ORIGINAL nonzero root36 square = 1296",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root36 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root36 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "36"
        ],
        "output": "1296",
        "label": "Quadratic ORIGINAL nonzero root36 square"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "ORIGINAL nonzero root x=36 in1..100;actual square=1296",
      "after": "Euclidean square=12*101+84;nonzero square class=84;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Quadratic ORIGINAL nonzero root37 square := mul(37, 37)",
      "after": "Quadratic ORIGINAL nonzero root37 square = 1369",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root37 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root37 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "37",
          "37"
        ],
        "output": "1369",
        "label": "Quadratic ORIGINAL nonzero root37 square"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "ORIGINAL nonzero root x=37 in1..100;actual square=1369",
      "after": "Euclidean square=13*101+56;nonzero square class=56;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Quadratic ORIGINAL nonzero root38 square := mul(38, 38)",
      "after": "Quadratic ORIGINAL nonzero root38 square = 1444",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root38 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root38 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "38"
        ],
        "output": "1444",
        "label": "Quadratic ORIGINAL nonzero root38 square"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "ORIGINAL nonzero root x=38 in1..100;actual square=1444",
      "after": "Euclidean square=14*101+30;nonzero square class=30;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Quadratic ORIGINAL nonzero root39 square := mul(39, 39)",
      "after": "Quadratic ORIGINAL nonzero root39 square = 1521",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root39 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root39 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "39"
        ],
        "output": "1521",
        "label": "Quadratic ORIGINAL nonzero root39 square"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "ORIGINAL nonzero root x=39 in1..100;actual square=1521",
      "after": "Euclidean square=15*101+6;nonzero square class=6;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Quadratic ORIGINAL nonzero root40 square := mul(40, 40)",
      "after": "Quadratic ORIGINAL nonzero root40 square = 1600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root40 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root40 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "40"
        ],
        "output": "1600",
        "label": "Quadratic ORIGINAL nonzero root40 square"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "ORIGINAL nonzero root x=40 in1..100;actual square=1600",
      "after": "Euclidean square=15*101+85;nonzero square class=85;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Quadratic ORIGINAL nonzero root41 square := mul(41, 41)",
      "after": "Quadratic ORIGINAL nonzero root41 square = 1681",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root41 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root41 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "41",
          "41"
        ],
        "output": "1681",
        "label": "Quadratic ORIGINAL nonzero root41 square"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "ORIGINAL nonzero root x=41 in1..100;actual square=1681",
      "after": "Euclidean square=16*101+65;nonzero square class=65;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Quadratic ORIGINAL nonzero root42 square := mul(42, 42)",
      "after": "Quadratic ORIGINAL nonzero root42 square = 1764",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root42 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root42 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "42"
        ],
        "output": "1764",
        "label": "Quadratic ORIGINAL nonzero root42 square"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "ORIGINAL nonzero root x=42 in1..100;actual square=1764",
      "after": "Euclidean square=17*101+47;nonzero square class=47;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Quadratic ORIGINAL nonzero root43 square := mul(43, 43)",
      "after": "Quadratic ORIGINAL nonzero root43 square = 1849",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root43 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root43 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "43",
          "43"
        ],
        "output": "1849",
        "label": "Quadratic ORIGINAL nonzero root43 square"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "ORIGINAL nonzero root x=43 in1..100;actual square=1849",
      "after": "Euclidean square=18*101+31;nonzero square class=31;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Quadratic ORIGINAL nonzero root44 square := mul(44, 44)",
      "after": "Quadratic ORIGINAL nonzero root44 square = 1936",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root44 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root44 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "44",
          "44"
        ],
        "output": "1936",
        "label": "Quadratic ORIGINAL nonzero root44 square"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "ORIGINAL nonzero root x=44 in1..100;actual square=1936",
      "after": "Euclidean square=19*101+17;nonzero square class=17;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Quadratic ORIGINAL nonzero root45 square := mul(45, 45)",
      "after": "Quadratic ORIGINAL nonzero root45 square = 2025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root45 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root45 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "45",
          "45"
        ],
        "output": "2025",
        "label": "Quadratic ORIGINAL nonzero root45 square"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "ORIGINAL nonzero root x=45 in1..100;actual square=2025",
      "after": "Euclidean square=20*101+5;nonzero square class=5;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Quadratic ORIGINAL nonzero root46 square := mul(46, 46)",
      "after": "Quadratic ORIGINAL nonzero root46 square = 2116",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root46 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root46 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "46",
          "46"
        ],
        "output": "2116",
        "label": "Quadratic ORIGINAL nonzero root46 square"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "ORIGINAL nonzero root x=46 in1..100;actual square=2116",
      "after": "Euclidean square=20*101+96;nonzero square class=96;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Quadratic ORIGINAL nonzero root47 square := mul(47, 47)",
      "after": "Quadratic ORIGINAL nonzero root47 square = 2209",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root47 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root47 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "47"
        ],
        "output": "2209",
        "label": "Quadratic ORIGINAL nonzero root47 square"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "ORIGINAL nonzero root x=47 in1..100;actual square=2209",
      "after": "Euclidean square=21*101+88;nonzero square class=88;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Quadratic ORIGINAL nonzero root48 square := mul(48, 48)",
      "after": "Quadratic ORIGINAL nonzero root48 square = 2304",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root48 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root48 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "48"
        ],
        "output": "2304",
        "label": "Quadratic ORIGINAL nonzero root48 square"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "ORIGINAL nonzero root x=48 in1..100;actual square=2304",
      "after": "Euclidean square=22*101+82;nonzero square class=82;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Quadratic ORIGINAL nonzero root49 square := mul(49, 49)",
      "after": "Quadratic ORIGINAL nonzero root49 square = 2401",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root49 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root49 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "49",
          "49"
        ],
        "output": "2401",
        "label": "Quadratic ORIGINAL nonzero root49 square"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "ORIGINAL nonzero root x=49 in1..100;actual square=2401",
      "after": "Euclidean square=23*101+78;nonzero square class=78;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Quadratic ORIGINAL nonzero root50 square := mul(50, 50)",
      "after": "Quadratic ORIGINAL nonzero root50 square = 2500",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root50 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root50 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "50",
          "50"
        ],
        "output": "2500",
        "label": "Quadratic ORIGINAL nonzero root50 square"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "ORIGINAL nonzero root x=50 in1..100;actual square=2500",
      "after": "Euclidean square=24*101+76;nonzero square class=76;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Quadratic ORIGINAL nonzero root51 square := mul(51, 51)",
      "after": "Quadratic ORIGINAL nonzero root51 square = 2601",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root51 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root51 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "51",
          "51"
        ],
        "output": "2601",
        "label": "Quadratic ORIGINAL nonzero root51 square"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "ORIGINAL nonzero root x=51 in1..100;actual square=2601",
      "after": "Euclidean square=25*101+76;nonzero square class=76;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Quadratic ORIGINAL nonzero root52 square := mul(52, 52)",
      "after": "Quadratic ORIGINAL nonzero root52 square = 2704",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root52 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root52 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "52",
          "52"
        ],
        "output": "2704",
        "label": "Quadratic ORIGINAL nonzero root52 square"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "ORIGINAL nonzero root x=52 in1..100;actual square=2704",
      "after": "Euclidean square=26*101+78;nonzero square class=78;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Quadratic ORIGINAL nonzero root53 square := mul(53, 53)",
      "after": "Quadratic ORIGINAL nonzero root53 square = 2809",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root53 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root53 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "53",
          "53"
        ],
        "output": "2809",
        "label": "Quadratic ORIGINAL nonzero root53 square"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "ORIGINAL nonzero root x=53 in1..100;actual square=2809",
      "after": "Euclidean square=27*101+82;nonzero square class=82;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Quadratic ORIGINAL nonzero root54 square := mul(54, 54)",
      "after": "Quadratic ORIGINAL nonzero root54 square = 2916",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root54 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root54 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "54",
          "54"
        ],
        "output": "2916",
        "label": "Quadratic ORIGINAL nonzero root54 square"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "ORIGINAL nonzero root x=54 in1..100;actual square=2916",
      "after": "Euclidean square=28*101+88;nonzero square class=88;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Quadratic ORIGINAL nonzero root55 square := mul(55, 55)",
      "after": "Quadratic ORIGINAL nonzero root55 square = 3025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root55 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root55 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "55",
          "55"
        ],
        "output": "3025",
        "label": "Quadratic ORIGINAL nonzero root55 square"
      }
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "ORIGINAL nonzero root x=55 in1..100;actual square=3025",
      "after": "Euclidean square=29*101+96;nonzero square class=96;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "Quadratic ORIGINAL nonzero root56 square := mul(56, 56)",
      "after": "Quadratic ORIGINAL nonzero root56 square = 3136",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root56 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root56 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "56",
          "56"
        ],
        "output": "3136",
        "label": "Quadratic ORIGINAL nonzero root56 square"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "ORIGINAL nonzero root x=56 in1..100;actual square=3136",
      "after": "Euclidean square=31*101+5;nonzero square class=5;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "Quadratic ORIGINAL nonzero root57 square := mul(57, 57)",
      "after": "Quadratic ORIGINAL nonzero root57 square = 3249",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root57 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root57 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "57",
          "57"
        ],
        "output": "3249",
        "label": "Quadratic ORIGINAL nonzero root57 square"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "ORIGINAL nonzero root x=57 in1..100;actual square=3249",
      "after": "Euclidean square=32*101+17;nonzero square class=17;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "Quadratic ORIGINAL nonzero root58 square := mul(58, 58)",
      "after": "Quadratic ORIGINAL nonzero root58 square = 3364",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root58 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root58 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "58",
          "58"
        ],
        "output": "3364",
        "label": "Quadratic ORIGINAL nonzero root58 square"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "ORIGINAL nonzero root x=58 in1..100;actual square=3364",
      "after": "Euclidean square=33*101+31;nonzero square class=31;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "Quadratic ORIGINAL nonzero root59 square := mul(59, 59)",
      "after": "Quadratic ORIGINAL nonzero root59 square = 3481",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root59 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root59 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "59",
          "59"
        ],
        "output": "3481",
        "label": "Quadratic ORIGINAL nonzero root59 square"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "ORIGINAL nonzero root x=59 in1..100;actual square=3481",
      "after": "Euclidean square=34*101+47;nonzero square class=47;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "Quadratic ORIGINAL nonzero root60 square := mul(60, 60)",
      "after": "Quadratic ORIGINAL nonzero root60 square = 3600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root60 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root60 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "60",
          "60"
        ],
        "output": "3600",
        "label": "Quadratic ORIGINAL nonzero root60 square"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "ORIGINAL nonzero root x=60 in1..100;actual square=3600",
      "after": "Euclidean square=35*101+65;nonzero square class=65;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "Quadratic ORIGINAL nonzero root61 square := mul(61, 61)",
      "after": "Quadratic ORIGINAL nonzero root61 square = 3721",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root61 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root61 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "61",
          "61"
        ],
        "output": "3721",
        "label": "Quadratic ORIGINAL nonzero root61 square"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "ORIGINAL nonzero root x=61 in1..100;actual square=3721",
      "after": "Euclidean square=36*101+85;nonzero square class=85;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "Quadratic ORIGINAL nonzero root62 square := mul(62, 62)",
      "after": "Quadratic ORIGINAL nonzero root62 square = 3844",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root62 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root62 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "62",
          "62"
        ],
        "output": "3844",
        "label": "Quadratic ORIGINAL nonzero root62 square"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "ORIGINAL nonzero root x=62 in1..100;actual square=3844",
      "after": "Euclidean square=38*101+6;nonzero square class=6;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "Quadratic ORIGINAL nonzero root63 square := mul(63, 63)",
      "after": "Quadratic ORIGINAL nonzero root63 square = 3969",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root63 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root63 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "63",
          "63"
        ],
        "output": "3969",
        "label": "Quadratic ORIGINAL nonzero root63 square"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "ORIGINAL nonzero root x=63 in1..100;actual square=3969",
      "after": "Euclidean square=39*101+30;nonzero square class=30;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "Quadratic ORIGINAL nonzero root64 square := mul(64, 64)",
      "after": "Quadratic ORIGINAL nonzero root64 square = 4096",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root64 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root64 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "64",
          "64"
        ],
        "output": "4096",
        "label": "Quadratic ORIGINAL nonzero root64 square"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "ORIGINAL nonzero root x=64 in1..100;actual square=4096",
      "after": "Euclidean square=40*101+56;nonzero square class=56;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "Quadratic ORIGINAL nonzero root65 square := mul(65, 65)",
      "after": "Quadratic ORIGINAL nonzero root65 square = 4225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root65 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root65 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "65",
          "65"
        ],
        "output": "4225",
        "label": "Quadratic ORIGINAL nonzero root65 square"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "ORIGINAL nonzero root x=65 in1..100;actual square=4225",
      "after": "Euclidean square=41*101+84;nonzero square class=84;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "Quadratic ORIGINAL nonzero root66 square := mul(66, 66)",
      "after": "Quadratic ORIGINAL nonzero root66 square = 4356",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root66 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root66 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "66",
          "66"
        ],
        "output": "4356",
        "label": "Quadratic ORIGINAL nonzero root66 square"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "ORIGINAL nonzero root x=66 in1..100;actual square=4356",
      "after": "Euclidean square=43*101+13;nonzero square class=13;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "Quadratic ORIGINAL nonzero root67 square := mul(67, 67)",
      "after": "Quadratic ORIGINAL nonzero root67 square = 4489",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root67 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root67 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "67",
          "67"
        ],
        "output": "4489",
        "label": "Quadratic ORIGINAL nonzero root67 square"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "ORIGINAL nonzero root x=67 in1..100;actual square=4489",
      "after": "Euclidean square=44*101+45;nonzero square class=45;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "Quadratic ORIGINAL nonzero root68 square := mul(68, 68)",
      "after": "Quadratic ORIGINAL nonzero root68 square = 4624",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root68 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root68 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "68",
          "68"
        ],
        "output": "4624",
        "label": "Quadratic ORIGINAL nonzero root68 square"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "ORIGINAL nonzero root x=68 in1..100;actual square=4624",
      "after": "Euclidean square=45*101+79;nonzero square class=79;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "Quadratic ORIGINAL nonzero root69 square := mul(69, 69)",
      "after": "Quadratic ORIGINAL nonzero root69 square = 4761",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root69 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root69 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "69",
          "69"
        ],
        "output": "4761",
        "label": "Quadratic ORIGINAL nonzero root69 square"
      }
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "ORIGINAL nonzero root x=69 in1..100;actual square=4761",
      "after": "Euclidean square=47*101+14;nonzero square class=14;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "Quadratic ORIGINAL nonzero root70 square := mul(70, 70)",
      "after": "Quadratic ORIGINAL nonzero root70 square = 4900",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root70 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root70 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "70",
          "70"
        ],
        "output": "4900",
        "label": "Quadratic ORIGINAL nonzero root70 square"
      }
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "ORIGINAL nonzero root x=70 in1..100;actual square=4900",
      "after": "Euclidean square=48*101+52;nonzero square class=52;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "Quadratic ORIGINAL nonzero root71 square := mul(71, 71)",
      "after": "Quadratic ORIGINAL nonzero root71 square = 5041",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root71 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root71 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "71",
          "71"
        ],
        "output": "5041",
        "label": "Quadratic ORIGINAL nonzero root71 square"
      }
    },
    {
      "id": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "ORIGINAL nonzero root x=71 in1..100;actual square=5041",
      "after": "Euclidean square=49*101+92;nonzero square class=92;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "Quadratic ORIGINAL nonzero root72 square := mul(72, 72)",
      "after": "Quadratic ORIGINAL nonzero root72 square = 5184",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root72 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root72 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "72",
          "72"
        ],
        "output": "5184",
        "label": "Quadratic ORIGINAL nonzero root72 square"
      }
    },
    {
      "id": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "ORIGINAL nonzero root x=72 in1..100;actual square=5184",
      "after": "Euclidean square=51*101+33;nonzero square class=33;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "Quadratic ORIGINAL nonzero root73 square := mul(73, 73)",
      "after": "Quadratic ORIGINAL nonzero root73 square = 5329",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root73 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root73 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "73",
          "73"
        ],
        "output": "5329",
        "label": "Quadratic ORIGINAL nonzero root73 square"
      }
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "ORIGINAL nonzero root x=73 in1..100;actual square=5329",
      "after": "Euclidean square=52*101+77;nonzero square class=77;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "Quadratic ORIGINAL nonzero root74 square := mul(74, 74)",
      "after": "Quadratic ORIGINAL nonzero root74 square = 5476",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root74 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root74 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "74",
          "74"
        ],
        "output": "5476",
        "label": "Quadratic ORIGINAL nonzero root74 square"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "ORIGINAL nonzero root x=74 in1..100;actual square=5476",
      "after": "Euclidean square=54*101+22;nonzero square class=22;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "Quadratic ORIGINAL nonzero root75 square := mul(75, 75)",
      "after": "Quadratic ORIGINAL nonzero root75 square = 5625",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root75 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root75 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "75",
          "75"
        ],
        "output": "5625",
        "label": "Quadratic ORIGINAL nonzero root75 square"
      }
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "ORIGINAL nonzero root x=75 in1..100;actual square=5625",
      "after": "Euclidean square=55*101+70;nonzero square class=70;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "Quadratic ORIGINAL nonzero root76 square := mul(76, 76)",
      "after": "Quadratic ORIGINAL nonzero root76 square = 5776",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root76 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root76 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "76",
          "76"
        ],
        "output": "5776",
        "label": "Quadratic ORIGINAL nonzero root76 square"
      }
    },
    {
      "id": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "ORIGINAL nonzero root x=76 in1..100;actual square=5776",
      "after": "Euclidean square=57*101+19;nonzero square class=19;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "Quadratic ORIGINAL nonzero root77 square := mul(77, 77)",
      "after": "Quadratic ORIGINAL nonzero root77 square = 5929",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root77 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root77 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "77",
          "77"
        ],
        "output": "5929",
        "label": "Quadratic ORIGINAL nonzero root77 square"
      }
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "ORIGINAL nonzero root x=77 in1..100;actual square=5929",
      "after": "Euclidean square=58*101+71;nonzero square class=71;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "Quadratic ORIGINAL nonzero root78 square := mul(78, 78)",
      "after": "Quadratic ORIGINAL nonzero root78 square = 6084",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root78 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root78 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "78",
          "78"
        ],
        "output": "6084",
        "label": "Quadratic ORIGINAL nonzero root78 square"
      }
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "ORIGINAL nonzero root x=78 in1..100;actual square=6084",
      "after": "Euclidean square=60*101+24;nonzero square class=24;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "Quadratic ORIGINAL nonzero root79 square := mul(79, 79)",
      "after": "Quadratic ORIGINAL nonzero root79 square = 6241",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root79 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root79 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "79",
          "79"
        ],
        "output": "6241",
        "label": "Quadratic ORIGINAL nonzero root79 square"
      }
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "ORIGINAL nonzero root x=79 in1..100;actual square=6241",
      "after": "Euclidean square=61*101+80;nonzero square class=80;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "Quadratic ORIGINAL nonzero root80 square := mul(80, 80)",
      "after": "Quadratic ORIGINAL nonzero root80 square = 6400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root80 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root80 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "80",
          "80"
        ],
        "output": "6400",
        "label": "Quadratic ORIGINAL nonzero root80 square"
      }
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "ORIGINAL nonzero root x=80 in1..100;actual square=6400",
      "after": "Euclidean square=63*101+37;nonzero square class=37;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "Quadratic ORIGINAL nonzero root81 square := mul(81, 81)",
      "after": "Quadratic ORIGINAL nonzero root81 square = 6561",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root81 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root81 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "81",
          "81"
        ],
        "output": "6561",
        "label": "Quadratic ORIGINAL nonzero root81 square"
      }
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "ORIGINAL nonzero root x=81 in1..100;actual square=6561",
      "after": "Euclidean square=64*101+97;nonzero square class=97;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "Quadratic ORIGINAL nonzero root82 square := mul(82, 82)",
      "after": "Quadratic ORIGINAL nonzero root82 square = 6724",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root82 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root82 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "82",
          "82"
        ],
        "output": "6724",
        "label": "Quadratic ORIGINAL nonzero root82 square"
      }
    },
    {
      "id": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "ORIGINAL nonzero root x=82 in1..100;actual square=6724",
      "after": "Euclidean square=66*101+58;nonzero square class=58;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "Quadratic ORIGINAL nonzero root83 square := mul(83, 83)",
      "after": "Quadratic ORIGINAL nonzero root83 square = 6889",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root83 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root83 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "83",
          "83"
        ],
        "output": "6889",
        "label": "Quadratic ORIGINAL nonzero root83 square"
      }
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "ORIGINAL nonzero root x=83 in1..100;actual square=6889",
      "after": "Euclidean square=68*101+21;nonzero square class=21;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "Quadratic ORIGINAL nonzero root84 square := mul(84, 84)",
      "after": "Quadratic ORIGINAL nonzero root84 square = 7056",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root84 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root84 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "84",
          "84"
        ],
        "output": "7056",
        "label": "Quadratic ORIGINAL nonzero root84 square"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "ORIGINAL nonzero root x=84 in1..100;actual square=7056",
      "after": "Euclidean square=69*101+87;nonzero square class=87;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "Quadratic ORIGINAL nonzero root85 square := mul(85, 85)",
      "after": "Quadratic ORIGINAL nonzero root85 square = 7225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root85 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root85 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "85",
          "85"
        ],
        "output": "7225",
        "label": "Quadratic ORIGINAL nonzero root85 square"
      }
    },
    {
      "id": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "ORIGINAL nonzero root x=85 in1..100;actual square=7225",
      "after": "Euclidean square=71*101+54;nonzero square class=54;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "Quadratic ORIGINAL nonzero root86 square := mul(86, 86)",
      "after": "Quadratic ORIGINAL nonzero root86 square = 7396",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root86 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root86 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "86",
          "86"
        ],
        "output": "7396",
        "label": "Quadratic ORIGINAL nonzero root86 square"
      }
    },
    {
      "id": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "ORIGINAL nonzero root x=86 in1..100;actual square=7396",
      "after": "Euclidean square=73*101+23;nonzero square class=23;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "Quadratic ORIGINAL nonzero root87 square := mul(87, 87)",
      "after": "Quadratic ORIGINAL nonzero root87 square = 7569",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root87 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root87 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "87",
          "87"
        ],
        "output": "7569",
        "label": "Quadratic ORIGINAL nonzero root87 square"
      }
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "ORIGINAL nonzero root x=87 in1..100;actual square=7569",
      "after": "Euclidean square=74*101+95;nonzero square class=95;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "Quadratic ORIGINAL nonzero root88 square := mul(88, 88)",
      "after": "Quadratic ORIGINAL nonzero root88 square = 7744",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root88 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root88 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "88",
          "88"
        ],
        "output": "7744",
        "label": "Quadratic ORIGINAL nonzero root88 square"
      }
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "ORIGINAL nonzero root x=88 in1..100;actual square=7744",
      "after": "Euclidean square=76*101+68;nonzero square class=68;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "Quadratic ORIGINAL nonzero root89 square := mul(89, 89)",
      "after": "Quadratic ORIGINAL nonzero root89 square = 7921",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root89 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root89 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "89",
          "89"
        ],
        "output": "7921",
        "label": "Quadratic ORIGINAL nonzero root89 square"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "ORIGINAL nonzero root x=89 in1..100;actual square=7921",
      "after": "Euclidean square=78*101+43;nonzero square class=43;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "Quadratic ORIGINAL nonzero root90 square := mul(90, 90)",
      "after": "Quadratic ORIGINAL nonzero root90 square = 8100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root90 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root90 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "90",
          "90"
        ],
        "output": "8100",
        "label": "Quadratic ORIGINAL nonzero root90 square"
      }
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "ORIGINAL nonzero root x=90 in1..100;actual square=8100",
      "after": "Euclidean square=80*101+20;nonzero square class=20;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "Quadratic ORIGINAL nonzero root91 square := mul(91, 91)",
      "after": "Quadratic ORIGINAL nonzero root91 square = 8281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root91 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root91 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "91",
          "91"
        ],
        "output": "8281",
        "label": "Quadratic ORIGINAL nonzero root91 square"
      }
    },
    {
      "id": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "ORIGINAL nonzero root x=91 in1..100;actual square=8281",
      "after": "Euclidean square=81*101+100;nonzero square class=100;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "Quadratic ORIGINAL nonzero root92 square := mul(92, 92)",
      "after": "Quadratic ORIGINAL nonzero root92 square = 8464",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root92 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root92 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "92",
          "92"
        ],
        "output": "8464",
        "label": "Quadratic ORIGINAL nonzero root92 square"
      }
    },
    {
      "id": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "ORIGINAL nonzero root x=92 in1..100;actual square=8464",
      "after": "Euclidean square=83*101+81;nonzero square class=81;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "Quadratic ORIGINAL nonzero root93 square := mul(93, 93)",
      "after": "Quadratic ORIGINAL nonzero root93 square = 8649",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root93 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root93 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "93",
          "93"
        ],
        "output": "8649",
        "label": "Quadratic ORIGINAL nonzero root93 square"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "ORIGINAL nonzero root x=93 in1..100;actual square=8649",
      "after": "Euclidean square=85*101+64;nonzero square class=64;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "Quadratic ORIGINAL nonzero root94 square := mul(94, 94)",
      "after": "Quadratic ORIGINAL nonzero root94 square = 8836",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root94 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root94 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "94",
          "94"
        ],
        "output": "8836",
        "label": "Quadratic ORIGINAL nonzero root94 square"
      }
    },
    {
      "id": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "ORIGINAL nonzero root x=94 in1..100;actual square=8836",
      "after": "Euclidean square=87*101+49;nonzero square class=49;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "Quadratic ORIGINAL nonzero root95 square := mul(95, 95)",
      "after": "Quadratic ORIGINAL nonzero root95 square = 9025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root95 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root95 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "95",
          "95"
        ],
        "output": "9025",
        "label": "Quadratic ORIGINAL nonzero root95 square"
      }
    },
    {
      "id": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "ORIGINAL nonzero root x=95 in1..100;actual square=9025",
      "after": "Euclidean square=89*101+36;nonzero square class=36;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "Quadratic ORIGINAL nonzero root96 square := mul(96, 96)",
      "after": "Quadratic ORIGINAL nonzero root96 square = 9216",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root96 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root96 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "96",
          "96"
        ],
        "output": "9216",
        "label": "Quadratic ORIGINAL nonzero root96 square"
      }
    },
    {
      "id": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "ORIGINAL nonzero root x=96 in1..100;actual square=9216",
      "after": "Euclidean square=91*101+25;nonzero square class=25;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "Quadratic ORIGINAL nonzero root97 square := mul(97, 97)",
      "after": "Quadratic ORIGINAL nonzero root97 square = 9409",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root97 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root97 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "97",
          "97"
        ],
        "output": "9409",
        "label": "Quadratic ORIGINAL nonzero root97 square"
      }
    },
    {
      "id": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "ORIGINAL nonzero root x=97 in1..100;actual square=9409",
      "after": "Euclidean square=93*101+16;nonzero square class=16;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "Quadratic ORIGINAL nonzero root98 square := mul(98, 98)",
      "after": "Quadratic ORIGINAL nonzero root98 square = 9604",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root98 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root98 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "98",
          "98"
        ],
        "output": "9604",
        "label": "Quadratic ORIGINAL nonzero root98 square"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "ORIGINAL nonzero root x=98 in1..100;actual square=9604",
      "after": "Euclidean square=95*101+9;nonzero square class=9;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "Quadratic ORIGINAL nonzero root99 square := mul(99, 99)",
      "after": "Quadratic ORIGINAL nonzero root99 square = 9801",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root99 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root99 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "99",
          "99"
        ],
        "output": "9801",
        "label": "Quadratic ORIGINAL nonzero root99 square"
      }
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "ORIGINAL nonzero root x=99 in1..100;actual square=9801",
      "after": "Euclidean square=97*101+4;nonzero square class=4;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "Quadratic ORIGINAL nonzero root100 square := mul(100, 100)",
      "after": "Quadratic ORIGINAL nonzero root100 square = 10000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL nonzero root100 square，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL nonzero root100 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "100",
          "100"
        ],
        "output": "10000",
        "label": "Quadratic ORIGINAL nonzero root100 square"
      }
    },
    {
      "id": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "ORIGINAL nonzero root x=100 in1..100;actual square=10000",
      "after": "Euclidean square=99*101+1;nonzero square class=1;complete root-index order retained",
      "relation": "implies",
      "rule_id": "quadratic_ALL_nonzero_square_rows_not_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐每個非零root，列真平方與mod分解；prime下非零x不會平方成0。不是只抽幾個residues或預用Euler／Legendre。",
        "en": "Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)",
      "after": "Quadratic distinct nonzero residue count = 50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic distinct nonzero residue count，不以小數近似替代。",
          "en": "Recompute Quadratic distinct nonzero residue count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1"
        ],
        "output": "50",
        "label": "Quadratic distinct nonzero residue count"
      }
    },
    {
      "id": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "Quadratic ORIGINAL modulus minus1 := sub(101, 1)",
      "after": "Quadratic ORIGINAL modulus minus1 = 100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic ORIGINAL modulus minus1，不以小數近似替代。",
          "en": "Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "101",
          "1"
        ],
        "output": "100",
        "label": "Quadratic ORIGINAL modulus minus1"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "Quadratic half of all nonzero roots := div(100, 2)",
      "after": "Quadratic half of all nonzero roots = 50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。",
        "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Quadratic half of all nonzero roots，不以小數近似替代。",
          "en": "Recompute Quadratic half of all nonzero roots with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "100",
          "2"
        ],
        "output": "50",
        "label": "Quadratic half of all nonzero roots"
      }
    },
    {
      "id": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "ORIGINAL ODD PRIME p=101;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]];distinct classes=[1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100]",
      "after": "x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=50=(101-1)/2=50",
      "relation": "implies",
      "rule_id": "quadratic_prime_field_opposite_root_pairs_exactly_two",
      "conditions": [],
      "explanation": {
        "zh-Hant": "prime給無零因子的乘積消去，odd保2x不等0；±配對互不重疊覆蓋全部p−1非零root。這證計數與兩根性，不外推到zero、p2或composite。",
        "en": "The prime-field no-zero-divisor property gives x=±y;oddness makes2x nonzero.The opposite pairs partition all p−1 nonzero roots.This proves both counting and two-root statements,not their extension to zero,2 or composite moduli."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      101,
      [
        [
          1,
          1
        ],
        [
          2,
          4
        ],
        [
          3,
          9
        ],
        [
          4,
          16
        ],
        [
          5,
          25
        ],
        [
          6,
          36
        ],
        [
          7,
          49
        ],
        [
          8,
          64
        ],
        [
          9,
          81
        ],
        [
          10,
          100
        ],
        [
          11,
          20
        ],
        [
          12,
          43
        ],
        [
          13,
          68
        ],
        [
          14,
          95
        ],
        [
          15,
          23
        ],
        [
          16,
          54
        ],
        [
          17,
          87
        ],
        [
          18,
          21
        ],
        [
          19,
          58
        ],
        [
          20,
          97
        ],
        [
          21,
          37
        ],
        [
          22,
          80
        ],
        [
          23,
          24
        ],
        [
          24,
          71
        ],
        [
          25,
          19
        ],
        [
          26,
          70
        ],
        [
          27,
          22
        ],
        [
          28,
          77
        ],
        [
          29,
          33
        ],
        [
          30,
          92
        ],
        [
          31,
          52
        ],
        [
          32,
          14
        ],
        [
          33,
          79
        ],
        [
          34,
          45
        ],
        [
          35,
          13
        ],
        [
          36,
          84
        ],
        [
          37,
          56
        ],
        [
          38,
          30
        ],
        [
          39,
          6
        ],
        [
          40,
          85
        ],
        [
          41,
          65
        ],
        [
          42,
          47
        ],
        [
          43,
          31
        ],
        [
          44,
          17
        ],
        [
          45,
          5
        ],
        [
          46,
          96
        ],
        [
          47,
          88
        ],
        [
          48,
          82
        ],
        [
          49,
          78
        ],
        [
          50,
          76
        ],
        [
          51,
          76
        ],
        [
          52,
          78
        ],
        [
          53,
          82
        ],
        [
          54,
          88
        ],
        [
          55,
          96
        ],
        [
          56,
          5
        ],
        [
          57,
          17
        ],
        [
          58,
          31
        ],
        [
          59,
          47
        ],
        [
          60,
          65
        ],
        [
          61,
          85
        ],
        [
          62,
          6
        ],
        [
          63,
          30
        ],
        [
          64,
          56
        ],
        [
          65,
          84
        ],
        [
          66,
          13
        ],
        [
          67,
          45
        ],
        [
          68,
          79
        ],
        [
          69,
          14
        ],
        [
          70,
          52
        ],
        [
          71,
          92
        ],
        [
          72,
          33
        ],
        [
          73,
          77
        ],
        [
          74,
          22
        ],
        [
          75,
          70
        ],
        [
          76,
          19
        ],
        [
          77,
          71
        ],
        [
          78,
          24
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        [
          79,
          80
        ],
        [
          80,
          37
        ],
        [
          81,
          97
        ],
        [
          82,
          58
        ],
        [
          83,
          21
        ],
        [
          84,
          87
        ],
        [
          85,
          54
        ],
        [
          86,
          23
        ],
        [
          87,
          95
        ],
        [
          88,
          68
        ],
        [
          89,
          43
        ],
        [
          90,
          20
        ],
        [
          91,
          100
        ],
        [
          92,
          81
        ],
        [
          93,
          64
        ],
        [
          94,
          49
        ],
        [
          95,
          36
        ],
        [
          96,
          25
        ],
        [
          97,
          16
        ],
        [
          98,
          9
        ],
        [
          99,
          4
        ],
        [
          100,
          1
        ]
      ],
      [
        1,
        4,
        5,
        6,
        9,
        13,
        14,
        16,
        17,
        19,
        20,
        21,
        22,
        23,
        24,
        25,
        30,
        31,
        33,
        36,
        37,
        43,
        45,
        47,
        49,
        52,
        54,
        56,
        58,
        64,
        65,
        68,
        70,
        71,
        76,
        77,
        78,
        79,
        80,
        81,
        82,
        84,
        85,
        87,
        88,
        92,
        95,
        96,
        97,
        100
      ],
      50,
      50,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, 是)",
    "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-20261111-r06-quadratic-residue-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r06-quadratic-residue-foundations/questions.jsonl",
      "batch_sha256": "91ac41444413e6e726b0226c78523599572fbb0fb658af56f719abb031599107",
      "original_line": 25,
      "original_line_sha256": "2b66bd6b089a34dbed7bea1d3eb3c9425088b7ab0f36a87654699ccedd366604",
      "question_compact_sha256": "831c7dc9730e28dfb5a264c5c0072a14fd27584d10cb1ceadb8b2fcc6a32a8a2",
      "english_row_compact_sha256": "1f9ec0023d145478e8571e551132f274f7ed2c1399a363769726762634bf6301",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r06-quadratic-residue-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r06-quadratic-residue-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-quadratic-residue": {
        "sha256": "c93521b58b1876da85beb3308b403a1cd27209001660dfc6c04815276f035e21",
        "english_sha256": "2b942134468bcf2df5c61739cc29561a0f40e3257048b11a8bc08a60de227a4d",
        "original_path": "public/data/mko/quadratic-residue.json",
        "english_path": "public/data/i18n/editions/mko-quadratic-residue.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8",
    "original_candidate_sha256": "c0a16186f4c78c9c2d2345234ad83859d92f6ccd23e87c02d97e172c898488c3",
    "original_candidate_line_sha256": "c0a16186f4c78c9c2d2345234ad83859d92f6ccd23e87c02d97e172c898488c3"
  },
  "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-20261026-009-p46",
    "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": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438",
      "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 206,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "31f652b68bb8387356aa688517595af989e7272c2b44dd1f191e478e6460e6fe",
      "arithmetic_steps": 103
    },
    "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."
  }
}